APPROVED: Anne V. Shepler, Major Professor Charles H. Conley, Committee Member Neal Brand, Committee Member Su Gao, Chair of the Department of

Mathematics Mark Wardell, Dean of the Toulouse

Graduate School

HOCHSCHILD COHOMOLOGY AND COMPLEX REFLECTION GROUPS

Briana A. Foster-Greenwood

Dissertation Prepared for the Degree of

DOCTOR OF PHILOSOPHY

UNIVERSITY OF NORTH TEXAS

August 2012

Foster-Greenwood, Briana A., Hochschild cohomology and complex reflection

groups. Doctor of Philosophy (Mathematics), August 2012, 88 pp., 3 tables, 2 figures,

bibliography, 28 titles.

A concrete description of Hochschild cohomology is the first step toward

exploring associative deformations of algebras. In this dissertation, deformation theory,

geometry, combinatorics, invariant theory, representation theory, and homological

algebra merge in an investigation of Hochschild cohomology of skew group algebras

arising from complex reflection groups. Given a linear action of a finite group on a

finite dimensional vector space, the skew group algebra under consideration is the semi-

direct product of the group with a polynomial ring on the vector space.

Each representation of a group defines a different skew group algebra, which may

have its own interesting deformations. In this work, we explicitly describe all graded

Hecke algebras arising as deformations of the skew group algebra of any finite group

acting by the regular representation. We then focus on rank two exceptional complex

reflection groups acting by any irreducible representation. We consider in-depth the

reflection representation and a nonfaithful rotation representation. Alongside our study

of cohomology for the rotation representation, we develop techniques valid for arbitrary

finite groups acting by a representation with a central kernel.

Additionally, we consider combinatorial questions about reflection length and

codimension orderings on complex reflection groups. We give algorithms using character

theory to compute reflection length, atoms, and poset relations. Using a mixture of

theory, explicit examples, and calculations using the software GAP, we show that

Coxeter groups and the infinite family G(m,1,n) are the only irreducible complex

reflection groups for which the reflection length and codimension orders coincide. We

describe the atoms in the codimension order for the groups G(m,p,n). For arbitrary

finite groups, we show that the codimension atoms are contained in the support of every

generating set for cohomology, thus yielding information about the degrees of generators

for cohomology.

Copyright 2012

by

Briana A. Foster-Greenwood

ii

ACKNOWLEDGMENTS

I thank my advisor Anne Shepler for patiently guiding me. I am greatful for her intuitive

explanations (blobs and arrows and squigglies included) and ability to imagine exciting

things where others might only see a rough gray pebble. I owe gratitude to Professors

Charles Conley, Matt Douglass, Joseph Kung, and Neal Brand who taught me at various

stages and have been very supportive throughout. I also thank Ray Beaulieu and Paul

Lewis who encouraged me early on before I ever chose a major and William Cherry for the

geometry course that inspired me to continue on to graduate school.

My friends and family also made it possible to complete this stage of the journey. Tushar

has been a patient and inquiring listener—challenging my understanding of everything and

providing comic relief whenever required. Our weekend seminars were when most of the

learning happened, as we shared mathematical “cookies” and explored recurring themes

among a miscellany of mathematics, whatever caught our attention that week. Christine

has been a source of energetic encouragement when I needed it most. My parents and

brother are always there to provide respite and perspective.

iii

CONTENTS

ACKNOWLEDGMENTS iii

LIST OF TABLES vii

LIST OF FIGURES viii

CHAPTER 1. INTRODUCTION 1

1.1. Complex Reflection Groups 2

1.2. Skew Group Algebras 3

1.3. Deformation Theory and Hochschild Cohomology 4

1.4. Motivation 5

1.5. Problems 7

1.5.1. Reflection Length and Codimension Posets for Complex Reflection Groups 7

1.5.2. Determination of Hochschild Cohomology 8

CHAPTER 2. COMPARING REFLECTION LENGTH AND CODIMENSION 10

2.1. Reflection Length and Codimension Posets 10

2.2. Functions versus Posets 13

2.3. The Infinite Family G(m, p, n) 14

2.4. Rank Two Exceptional Reflection Groups 18

2.5. Exceptional Reflection Groups G23 −G37 20

2.6. When Does Reflection Length Equal Codimension? 22

2.7. Applications to Hochschild Cohomology of Skew Group Algebras 23

CHAPTER 3. TECHNIQUES FOR COMPUTING HOCHSCHILD COHOMOLOGY

OF SKEW GROUP ALGEBRAS 28

iv

3.1. Invariant Theory 28

3.2. Actions of Centralizers on Fixed Point Spaces 29

3.3. Poincare Series 29

3.4. Necessary Conditions for Nonzero Cohomology 31

3.5. Cohomology Component for an Element with an Abelian Centralizer 31

3.6. Cohomology Component for an Element with a Trivial Fixed Point Space 33

3.7. Galois Conjugate Representations 34

3.8. Representations Differing by a Group Automorphism 34

CHAPTER 4. RANK TWO EXCEPTIONAL REFLECTION GROUPS 36

4.1. Centralizer Subgroups of Reflection Groups G4 −G22 36

4.2. Cohomology for Irreducible Representations 37

4.3. Cohomology for the Reflection Representation 39

CHAPTER 5. COHOMOLOGY FOR LIFTS OF QUOTIENT GROUP

REPRESENTATIONS 41

5.1. The Lazy Mathematician’s Dream Formula 41

5.2. The Rotation Representation 45

5.3. Cohomology for the Alternating Group on Four Letters 45

5.3.1. Identity Component 46

5.3.2. Double-flip Component 51

5.3.3. Three-cycle Components 52

5.4. Cohomology for Tetrahedral Groups Acting by the Rotation Representation 53

CHAPTER 6. REGULAR REPRESENTATION 56

6.1. G-invariant Two-forms 57

6.2. Polynomial Degree Zero Hochschild 2-cohomology 59

6.3. From Cohomology to Graded Hecke Algebras 62

APPENDIX A. REFLECTION LENGTH AND CODIMENSION CODE

66

v

APPENDIX B. POINCARE POLYNOMIALS FOR ABSOLUTE REFLECTION

LENGTH AND CODIMENSION 79

APPENDIX C. HOCHSCHILD COHOMOLOGY TABLES FOR GROUP G6 82

BIBLIOGRAPHY 86

vi

LIST OF TABLES

2.1 Atom Count in (CG,≤⊥) for Exceptional Reflection Groups G23 −G37 22

B.1 Reflection Length and Codimension Polynomials for Groups G4 −G22 80

B.2 Reflection Length and Codimension Polynomials for Groups G23 −G37 81

vii

LIST OF FIGURES

2.1 Reflection Length and Codimension Posets for the Dihedral Group D4 12

2.2 Reflection Length and Codimension Posets for Reflection Group G4 13

viii

CHAPTER 1

INTRODUCTION

In this dissertation, we investigate cohomology governing deformations of algebras. To

illuminate the idea of a deformation, we begin with an informal example that has both a

geometric and algebraic interpretation. Consider the following curve in the plane:

Should we view the singularity (or sharp point) as an accident, or fundamental? We wonder:

Is the curve the shadow of a smooth curve in another dimension?

Or the curve obtained when a loop is pulled tight?

t = 0 t = 14

t = 12

t = 1

The family of loops above is an example of a deformation of the curve y2 = x3. The family

is parametrized—each loop has equation y2 = x3 + tx2, for some real value t. As t varies, so

do the curves, and the special case t = 0 gives the singular curve we are trying to deform.

At the same time, we could consider the coordinate rings

R[x, y, t]/〈y2 − x3 − tx2〉

of these curves, obtaining a parametrized family of algebras that may be regarded as a

deformation of the algebra R[x, y]/〈y2 − x3〉 (see [8]).

1

Formal deformations of algebras (see Section 1.3) do not require us to have any geometric

context in mind—they are defined for any associative algebra. Determining all possible

associative deformations of an algebra invokes a study of Hochschild cohomology.

In this work, the algebras of interest are skew group algebras built from groups and their

actions on space. Using invariant theory, representation theory, cohomology, combinatorics,

and geometry, we explore the Hochschild cohomology and deformation theory of skew group

algebras arising from different representations of reflection groups. Background definitions

and motivation are contained in Sections 1.1−1.4, and the problems of investigation are

outlined in Section 1.5.

1.1. Complex Reflection Groups

Let V be a real or complex n-dimensional vector space. An element of GL(V ) is a

reflection if it has finite order and its fixed point space is a hyperplane. Thus the eigenvalues

of a reflection are 1, . . . , 1, λ for some root of unity λ 6= 1. A reflection is called real or complex

according to whether V is a real or complex vector space. While a real reflection must have

order two, a complex reflection can have any finite order greater than one. A finite subgroup

of GL(V ) is a reflection group if it is generated by reflections. With respect to a suitable

basis, a finite reflection group acts by isometries, i.e., preserves a Hermitian form and is

represented by unitary matrices.

Examples of reflection groups include dihedral groups (acting on R2 as reflections and

rotations of planar n-gons) and symmetric groups (generated by transpositions fixing the

hyperplanes xi − xj = 0 in real or complex space). Finite irreducible complex reflection

groups were classified in the 1950’s by Shephard and Todd [21]. The classification consists

of one infinite family G(m, p, n) and 34 exceptional reflection groups, which are referred to

as G4 −G37 in the literature.

Reflection groups are remarkable for several reasons. They include symmetry groups of

the five Platonic solids in R3 as well as regular polytopes in complex space. The Shephard-

Todd-Chevalley Theorem [4] of classical invariant theory establishes reflection groups as

2

precisely those groups whose ring of polynomial invariants is a polynomial algebra (gener-

ated by algebraically independent invariant polynomials). In the realm of Lie theory, the

combinatorics of Weyl groups is elegantly connected to the classification of simple complex

Lie algebras. The action of a reflection group on the set of hyperplanes and also a set of root

vectors allows one to study reflection groups using geometry and combinatorics.

1.2. Skew Group Algebras

Let G be a finite group acting on a C-algebra A by algebra automorphisms (so, formally,

we have a group homomorphism from G to Aut(A)). The skew group algebra A#G is the

semi-direct product of A with the group algebra CG. As a vector space, A#G ∼= A ⊗ CG,

so every element of A#G can be expressed uniquely as a sum∑

g∈G fg ⊗ g with fg ∈ A.

Multiplying two simple tensors requires use of the group action to move a group element

past an element of A:

(a1 ⊗ g1)(a2 ⊗ g2) := a1 ~g1(a2)⊗ g1g2,

where ~g1(a2) is the result in A of applying the action of g1 to a2. This definition extends

bilinearly to an associative multiplication on all of A#G. Note that A#G contains both A

and CG as subalgebras. Furthermore, if G acts faithfully on a commutative algebra A, then

the skew group algebra A#G contains as its center the ring AG = {a ∈ A : ~g(a) = a} of

G-invariants. For notational convenience, we often supress the tensor signs when working

with a skew group algebra.

If G acts linearly on a vector space V ∼= Cn with basis v1, . . . , vn, then G also acts on the

symmetric algebra S(V ) ∼= C[v1, . . . , vn], and we can form the skew group algebra

S(V )#G := S(V ) oCG.

Within this algebra, vectors commute (vw − wv = 0 for all v, w in V ), group elements

multiply as they do in the group, and when a group element moves from left to right past a

vector, it acts on the vector. By changing the group action, we obtain a whole host of skew

group algebras for a given group.

3

1.2.1. Example. Let G be the cyclic group generated by the permutation (1 2 3). Then G

acts on V ∼= C3 by cyclically permuting basis vectors v1, v2, v3. Every element of S(V )#G

can be expressed uniquely in the form

f1 + f2(1 2 3) + f3(3 2 1),

with each fi a polynomial in C[v1, v2, v3]. To illustrate the multiplication in the skew group

algebra, consider the product of (1 2 3) and v21v72v

43(3 2 1):

(1 2 3)v21v72v

43(3 2 1) = v22v

73v

41(1 2 3)(3 2 1) = v41v

22v

73.

1.3. Deformation Theory and Hochschild Cohomology

In algebraic deformation theory (see [10], [13], and [8]), we seek to deform the multipli-

cation in an algebra while retaining the same underlying vector space structure. Formally,

we adjoin a central parameter t to an F-algebra A and consider the vector space A[t] with

multiplication defined by

a ∗ b = ab+ µ1(a, b)t+ µ2(a, b)t2 + · · · ,

where a, b are in A, and each µi : A × A → A is bilinear1. We obtain a family of algebras

by specializing the parameter t to different values in the field (e.g. letting t = 0 recovers the

algebra A).

Requiring an associative multiplication imposes conditions on the maps µi and leads to

a study of Hochschild cohomology. Indeed, expanding (a ∗ b) ∗ c and a ∗ (b ∗ c) into power

series and equating like terms produces a sequence of “associative deformation equations”,

the first of which provides the definition of a Hochschild 2-cocycle. In particular, the first

multiplication map µ1 : A × A → A of an associative deformation must satisfy the cocycle

condition

aµ1(b, c)− µ1(ab, c) + µ1(a, bc)− µ1(a, b)c = 0.

1The multiplication is defined first on A and then extended F[t]-bilinearly to all of A[t]. For each pair

a, b in A we require µi(a, b) = 0 for all but finitely many i ≥ 1 so that a ∗ b is in A[t] as opposed to A[[t]].

4

Generally, for a C-algebra A and an A-bimodule M , the Hochschild cohomology of A

with coefficients in M is the space

HH•(A,M) = Ext•A⊗Aop(A,M),

where we tensor over C. When M = A, we simply write HH•(A).

In the setting of skew group algebras, Hochschild cohomology may be formulated in terms

of invariant theory. Stefan [27] finds cohomology of the skew group algebra S(V )#G as

the space of G-invariants in a larger cohomology ring:

HH•(S(V )#G) ∼= HH•(S(V ), S(V )#G)G,

and Farinati [7] and Ginzburg and Kaledin [11] describe the larger cohomology ring:

HH•(S(V ), S(V )#G) ∼=⊕g∈G

(S(V g)⊗

•−codim(g)∧(V g)∗ ⊗

codim(g)∧ ((V g)∗

)⊥ ⊗ Cg).

Here, V g denotes the fixed point space of g, and codim(g) := codimV g. After switching

the order of the operations “direct sum” and “take invariants”, we compute cohomology by

finding invariants under centralizer subgroups and taking a direct sum over a set of conjugacy

class representatives.

1.4. Motivation

Let G be a finite group acting linearly on a vector space V ∼= Cn. Let T (V ) be the tensor

algebra (over C) of V . Any skew-symmetric bilinear map κ : V ×V → CG defines a quotient

algebra

Hκ := T (V )#G/Iκ, .

where Iκ is the two-sided ideal Iκ = 〈v ⊗ w − w ⊗ v − κ(v, w) : v, w ∈ V 〉. We say Hκ is

a graded Hecke algebra if it satisfies a Poincare-Birkhoff-Witt property—namely, if the

associated graded algebra gr(Hκ) is isomorphic to the skew group algebra S(V )#G. (This

property is in analogy with the Poincare-Birkhoff-Witt Theorem for universal enveloping

algebras.) More concretely, Hκ is a graded Hecke algebra if the cosets ve11 · · · venn g+ Iκ for ei

in N and g in G form a vector space basis of Hκ, where v1, . . . , vn is any basis of V .

5

1.4.1. Example. As in Example 1.2.1, let G be the cyclic group generated by the permuta-

tion (1 2 3), and let G act on V ∼= C3 by cyclically permuting basis vectors v1, v2, v3. Define

a skew-symmetric bilinear form κ : V × V → CG by

κ(v1, v2) = κ(v2, v3) = κ(v3, v1) = 1 + (1 2 3) + (3 2 1).

Then Hκ is a graded Hecke algebra, as we will show in Chapter 6.

Graded Hecke algebras appear in different guises under many different names. Though

Drinfeld [5] was first to consider the algebras Hκ as defined above, the terminology “graded

Hecke algebra” arises because Lusztig [16] independently explored the case when G is a Weyl

group, defining the algebras as very different looking quotient algebras that gave a graded

version of the affine Hecke algebra (see [18]). In the case of a doubled-up action of G on

a vector space V ⊕ V ∗ with V a real or complex reflection representation, graded Hecke

algebras are known as rational Cherednik algebras and play the key role in Gordon’s

proof [12] of a Weyl group analogue of the n!-conjecture. More generally, if G is a symplectic

group acting on an even-dimensional vector space, then graded Hecke algebras are called

symplectic reflection algebras—rediscovered by Etingof and Ginzburg [6] in the context

of orbifold theory.

What does all this have to do with deformation theory? Every graded Hecke algebra is a

formal deformation of some skew group algebra S(V )#G. In fact, Witherspoon [28] shows

that, up to isomorphism, graded Hecke algebras are precisely the deformations of S(V )#G

such that the ith multiplication map has degree−2i for each i. However, Ram and Shepler [18]

show that for most complex reflection groups acting on V via the reflection representation,

the skew group algebra S(V )#G has no nontrivial graded Hecke algebras. This prompted

the idea to consider skew group algebras arising from nonfaithful and nonreflection actions.

For example, Shepler and Witherspoon [23] investigate a nonfaithful action of G(m, p, n)

and find more graded Hecke algebras.

6

What other interesting deformations of S(V )#Gmay play a role in representation theory?

We turn to Hochschild cohomology to give a description of all possible deformations, not

just the type discussed above.

1.5. Problems

In this dissertation, we explore two problems aimed towards understanding Hochschild

cohomology and deformations of skew group algebras associated to reflection groups. In

Chapter 2, we investigate a combinatorial problem about reflection groups and apply our

results to obtain information about generators for cohomology rings. In Chapters 3−6, we

turn our attention to determining Hochschild cohomology for skew group algebras arising

from various nonreflection representations of reflection groups.

1.5.1. Reflection Length and Codimension Posets for Complex Reflection Groups

Given a reflection group, we consider two partial orders: a reflection length order (related

to the word metric of geometric group theory) and a codimension order (capturing the

geometry of the group action). Various reflection length orders are key tools in the theory of

Coxeter groups, while codimension appears in the numerology of complex reflection groups

and in recent work connecting combinatorics and Hochschild cohomology.

For Coxeter groups and the infinite family G(m, 1, n) of monomial reflection groups,

the reflection length and codimension orders are known to be identical (see Carter [2] and

Shi [26]). Using a mixture of theory, explicit examples, and algorithms developed using the

software GAP, my work examines the question of whether or not this pattern holds for other

reflection groups. We complete the determination of the reflection groups for which reflection

length coincides with codimension:

Theorem. Let G be an irreducible complex reflection group. Absolute reflection length and

codimension coincide if and only if G is a Coxeter group or G = G(m, 1, n).

This theorem has implications for generators of cohomology rings related to deformation

theory of skew group algebras. The Hochschild cohomology ring of a skew group algebra may

be identified with the G-invariant subalgebra of a larger cohomology ring, which Shepler and

7

Witherspoon [24] show is finitely generated (under cup-product) by elements corresponding

to atoms of the codimension poset. Reflections are always atoms in the codimension poset,

but are there any other atoms?

In the case of the infinite family G(m, p, n), we obtain an explicit combinatorial descrip-

tion of the atoms in the codimension poset. For the exceptional reflection groups, we appeal

to character theory to write code in the software GAP to compute absolute reflection length,

atoms, and poset relations in the partial orders on the set of conjugacy classes of G.

1.5.2. Determination of Hochschild Cohomology

Working with the invariant theoretic formulation of cohomology requires an analysis

of centralizer subgroups and computation of invariants in tensor product spaces. Using

theoretical tools (such as character theory and Molien series) alongside computational algebra

software (such as GAP and Mathematica) we determine Hochschild cohomology for the

following cases:

Irreducible representations of G4 −G22: Let G be one of the groups G4 − G22,

and let V be any irreducible representation of G. To compute Hochschild cohomol-

ogy of S(V )#G, we need to understand centralizer subgroups and actions on fixed

spaces (and their complements) before computing invariants in a tensor product.

Fortunately, the centralizers in the groups G4 −G22 are very small. If z generates

the center of the group, then the centralizer of a noncentral element g in G is

generated by z and one more element (often g itself). Since these centralizers are

abelian, much of the Hochschild cohomology can be expressed in terms of polyno-

mials semi-invariant with respect to various linear characters. Using the software

GAP, we implement code to compute the relevant linear characters.

Rotation representation of G4 −G22: The groups G4−G22 in the Shephard-Todd

classification of irreducible complex reflection groups are called tetrahedral, octahe-

dral, or icosahedral, according to whether the quotient by the center Z is isomorphic

to the rotation group of the tetrahedron, octahedron, or icosahedron. Let V ∼= C3.

8

We compare cohomology of S(V )#G with the cohomology of S(V )#G/Z, where

the action of G on V is obtained by lifting the natural “rotation representation” of

G/Z on V . Naively, one would hope to obtain the cohomology for S(V )#G by just

taking multiple copies of the cohomology for S(V )#G/Z, but the computation is

more subtle because the centralizer of an element g in G need not project onto the

full centralizer of the corresponding element g in G/Z.

Regular representation: In Chapter 6, we classify all graded Hecke algebras arising

as deformations of S(V )#G when G is a finite group acting on V ∼= C|G| by the

(left) regular representation. In the case of the regular representation, very few

elements contribute to the cohomology HH•(S(V )#G) in the low degrees that we

are interested in for deformation theory. When |G| > 4, we identify the parameter

space of graded Hecke algebras with the space of skew-symmetric, G-invariant,

bilinear forms on V ∼= C|G|. With respect to an appropriate basis, these forms may

be described explicitly using matrices for the right regular representation.

9

CHAPTER 2

COMPARING REFLECTION LENGTH AND CODIMENSION

Reflection length and codimension of fixed point spaces induce partial orders on a complex

reflection group. While these partial orders are of independent combinatorial interest, our

investigation is motivated by a connection between the codimension order and the algebraic

structure of cohomology governing deformations of skew group algebras. In this chapter,

we compare the reflection length and codimension functions and discuss implications for

cohomology of skew group algebras. We give algorithms using character theory for computing

reflection length, atoms, and poset relations. Using a mixture of theory, explicit examples,

and computer calculations in GAP, we show that Coxeter groups and the infinite family

G(m, 1, n) are the only irreducible complex reflection groups for which the reflection length

and codimension orders coincide. We describe the atoms in the codimension order for the

infinite family G(m, p, n), which immediately yields an explicit description of generators for

cohomology.

2.1. Reflection Length and Codimension Posets

Let V be an n-dimensional vector space over R or C, and let G ⊂ GL(V ) be a finite

group generated by reflections. We define two class functions on a reflection group G and

use these functions to partially order the group. The first is a reflection length function,

related to the word metric of geometric group theory:

2.1.1. Definition. The (absolute) reflection length of an element g of a reflection group

G is the minimum number of factors needed to write g as a product of reflections:

`(g) = min{k : g = s1 · · · sk for some reflections s1, . . . , sk in G}.

The identity is declared to have length zero.

10

Note that the absolute reflection length function gives length with respect to all reflec-

tions in the group, as opposed to a set of fundamental or simple reflections.

The second function of interest relates to the geometry of the group action. While each

reflection in G fixes a hyperplane pointwise, each remaining element fixes an intersection of

hyperplanes. The codimension function records codimensions of these fixed point spaces:

2.1.2. Definition. The codimension of an element g in a group G ⊂ GL(V ) is the codi-

mension of its fixed point space (or number of non-one eigenvalues):

codim(g) = n− dim{v ∈ V : gv = v}.

Note that the identity has codimension zero, and reflections have codimension one.

The reflection length and codimension functions satisfy the following properties:

• constant on conjugacy classes

• subadditive: `(ab) ≤ `(a) + `(b) and codim(ab) ≤ codim(a) + codim(b)

• codim(g) ≤ `(g) for all g in G.

Now define the reflection length order on G by

a ≤`c ⇔ `(a) + `(a−1c) = `(c).

Analogously1, define the codimension order on G by

a ≤⊥ c ⇔ codim(a) + codim(a−1c) = codim(c).

Since reflection length and codimension are constant on conjugacy classes, we get induced

partial orders on the set of conjugacy classes of G. Define the reflection length (likewise

codimension) of a conjugacy class to be the reflection length (likewise codimension) of the

elements in the conjugacy class. If A and C are conjugacy classes of G, then set A ≤`C if

there exists an element a ∈ A and an element c ∈ C with a ≤`c. Analogously, define the

1Brady and Watt [1] prove ≤⊥ is a partial order. Their proof is also valid when codimension is replaced

by any function µ : G → [0,∞) satisfying µ(a) = 0 iff a = 1 (positive definite) and µ(ab) ≤ µ(a) + µ(b) for

all a, b in G (subadditive).

11

codimension order on conjugacy classes. As an example, Figure 2.1 illustrates the reflection

length and codimension orderings on the set of conjugacy classes of the dihedral group of

order eight.

1

F,R2F RF,R3F

R2 R,R−1

Reflection length poset1

F,R2F RF,R3F

R2 R,R−1

Codimension poset

Figure 2.1. Reflection Length and Codimension Posets for the Dihedral

Group D4

In Section 2.5, we will appeal to character theory to deduce information about the partial

orders on G by working with the (somewhat simpler) partial orders on the set of conjugacy

classes of G.

In a poset (P,≤), we say b covers a if b > a and the interval {x ∈ P : a < x < b} is

empty. The atoms of a poset are the covers of the minimum element (when it exists). The

identity is the minimum element in the reflection length poset and in the codimension poset.

For emphasis, we often refer to the atoms in the codimension poset as codimension atoms.

Note that an element a in G is an atom in the poset on G if and only if its conjugacy class

is an atom in the corresponding poset on the set of conjugacy classes of G.

Figure 2.2 illustrates the reflection length and codimension orderings on the set of con-

jugacy classes of the order 24 complex reflection group G4. Note that the conjugacy classes

labeled 3a and 3b consist of order three reflections and are the only atoms in the reflection

length poset. However, there is an additional atom (conjugacy class 2a) in the codimension

poset.

12

1a

3a

6b

2a

3b

6a4a

Reflection length poset

1a

3a 3b

6b 4a 6a 2a

Codimension poset

Figure 2.2. Reflection Length and Codimension Posets for Reflection Group G4

2.2. Functions versus Posets

We now show that comparing the length and codimension functions is (in a sense) equiv-

alent to comparing the set of atoms in each poset.

2.2.1. Definition. We say g = g1 · · · gk is a factorization of g with codimensions

adding if codim(g) = codim(g1) + · · ·+ codim(gk).

Note that if g = g1 · · · gk is a factorization with codimensions adding, then, using the fact that

codimension is subadditive and constant on conjugacy classes, we also have g1, . . . , gk ≤⊥ g.

Furthermore, since V is finite dimensional, we can work recursively to factor any nonidentity

element of G into a product of codimension atoms with codimensions adding:

2.2.2. Observation. Given a nonidentity group element g, there exist codimension atoms

a1, . . . , ak ≤⊥ g such that g = a1 · · · ak and codim(g) = codim(a1) + · · ·+ codim(ak).

The next two lemmas follow from repeated use of subadditivity of length and codimension

and the fact that codimension is bounded above by reflection length.

2.2.3. Lemma. Fix g in G. If `(a) = codim(a) for every codimension atom a ≤⊥ g, then

`(g) = codim(g).

Proof. The statement certainly holds for the identity. Now let g = a1 · · · ak be a factoriza-

tion of g 6= 1 into atoms with codimensions adding. (Note that necessarily a1, . . . , ak ≤⊥ g.)

13

Then

`(g) ≤ `(a1) + · · ·+ `(ak) = codim(a1) + · · ·+ codim(ak) = codim(g) ≤ `(g),

with equality throughout. �

2.2.4. Lemma. Let g ∈ G with `(g) = codim(g). If h ≤`g, then h ≤⊥ g.

Proof. Subadditivity gives codim(g) ≤ codim(h) + codim(h−1g). If `(g) = codim(g) and

h ≤`g, we also have the reverse inequality:

codim(h) + codim(h−1g) ≤ `(h) + `(h−1g) = `(g) = codim(g).

�

Lemma 2.2.3 and Lemma 2.2.4 combine to reveal that the reflection length and codimen-

sion functions coincide on all of G if and only if every codimension atom is a reflection.

2.2.5. Proposition. The following are equivalent:

(i) `(g) = codim(g) for every g in G.

(ii) `(g) = codim(g) for every codimension atom g in G.

(iii) Every codimension atom is a reflection.

(iv) For every g 6= 1, there exists a reflection s in G such that codim(gs) < codim(g).

Proof. The implication (1) ⇒ (2) is immediate. Application of Lemma 2.2.4 with g a

codimension atom and h a reflection shows (2) ⇒ (3). Lastly, if every codimension atom is

a reflection, then the hypothesis of Lemma 2.2.3 holds for each g in G, and hence (3)⇒ (1).

It is straightforward to work (4) into the loop via (1) ⇒ (4) and (4) ⇒ (3). �

2.3. The Infinite Family G(m, p, n)

The group G(m, 1, n) ∼= (Z/mZ)noSn consists of all n×n monomial matrices having mth

roots of unity for the nonzero entries. For p dividing m, the group G(m, p, n) is the subgroup

of G(m, 1, n) consisting of those elements whose nonzero entries multiply to an (mp

)th root

of unity. Throughout this section let ζm = e2πi/m. Each group G(m, p, n) contains the order

14

two transposition type reflections of the form δσ, where σ is a transposition swapping the ith

and jth basis vectors, and δ = diag(1, . . . , ζam, . . . , ζ−am , . . . , 1) scales rows i and j of σ (δ = 1

is a possibility). When p properly divides m, the group G(m, p, n) also contains the diagonal

reflections diag(1, . . . , ζam, . . . , 1) where 0 < a < m and p divides a. The G(m, p, n) family

includes the following Coxeter groups (n ≥ 2):

• symmetric group: G(1, 1, n) (not irreducible)

• Weyl groups of type Bn and Cn: G(2, 1, n)

• Weyl groups of type Dn: G(2, 2, n)

• dihedral groups: G(m,m, 2)

In this section, we describe the atoms in the codimension poset for an arbitrary group

G(m, p, n). In the groups for which the reflection length and codimension functions do not

coincide, we give explicit examples of elements with length exceeding codimension.

2.3.1. Definition. Let V = V1⊕· · ·⊕Vn be a decomposition of V ∼= Cn into one-dimensional

subspaces permuted by G(m, p, n). Let g be in G(m, p, n), and partition {V1, . . . , Vn} into

g-orbits, say O1, . . . ,Or. The action of g on⊕

Vj∈Oi Vj can be expressed as δiσi, where δi is

diagonal and σi is a cyclic permutation. (Thus, up to conjugation by a permutation matrix,

g is block diagonal with ith block δiσi.) The cycle-sum of g corresponding to orbit Oi is

the exponent ci (well-defined modulo m) such that det(δi) = ζcim.

Cycle-sums allow us to quickly read off codimension of an element:

codim(g) = n−#{i : ci ≡ 0 (mod m)}.

Note that for a reflection t and any group element g, the relation t ≤⊥ g is equivalent to

codim(t−1g) = codim(g)−1. Letting s = t−1 and noting that the conjugate elements sg and

gs have the same codimension, we obtain the following convenient observation:

2.3.2. Observation. An element g 6= 1 is comparable with a reflection in the codimension

poset if and only if there exists a reflection s such that codim(gs) < codim(g).

15

We recall from Shi [26] (see Corollary 1.8 and the proof of Theorem 2.1) the three

possibilities for how the cycle-sums change upon multiplying by a reflection:

2.3.3. Lemma (Shi [26]). Let g ∈ G(m, p, n) with cycle-sums c1, . . . , cr corresponding to g-

orbits O1, . . . ,Or. If s is a transposition type reflection interchanging Vi and Vj, then the

cycle-sums of g split or merge into the cycle-sums of gs:

(i) If Vi and Vj are in the same g-orbit, say Ok, then gs has cycle sums

c1, . . . , ck, . . . , cr, d, ck − d for some integer d.

(ii) If Vi and Vj are in different g-orbits, say Ok and Ol, then gs has cycle sums

c1, . . . , ck, . . . , cl, . . . , cr, ck + cl.

Let s be a diagonal reflection scaling Vi by non-1 eigenvalue ζam (where p divides a).

(iii) If Vi is in the g-orbit Ok, then gs has cycle sums c1, . . . , ck + a, . . . , cr.

Note that if g is nondiagonal, then by choosing a suitable transposition type reflection s,

we can arrange for the cycle-sum d of gs in part (1) of Lemma 2.3.3 to be any of 0, . . . ,m− 1.

In particular, we can choose s so that d = 0, thereby increasing the number of zero cycle-

sums and decreasing codimension. Hence every nonreflection atom in the codimension poset

must be diagonal. The converse is false, but we come closer to the set of nonreflection atoms

by considering only p-connected diagonal elements.

2.3.4. Definition. A diagonal matrix g 6= 1 whose non-1 eigenvalues are ζc1m , . . . , ζckm (listed

with multiplicities) is p-connected if p divides c1 + · · · + ck but p does not divide∑

i∈I ci

for I ( {1, . . . , k}. (Note that g is in G(m, p, n) iff p divides c1 + · · ·+ ck.)

It is easy to see that each nonidentity diagonal element of G(m, p, n) factors in G(m, p, n)

into p-connected elements with codimensions adding. Thus every nonreflection atom in

the codimension poset must be p-connected. We next check for poset relations among the

reflections and p-connected elements.

16

2.3.5. Lemma. The p-connected elements of G(m, p, n) are pairwise incomparable in the

codimension poset.

Proof. Suppose a, b in G(m, p, n) are diagonal elements such that ab is p-connected and

codim(a) + codim(b) = codim(ab). Since codimensions add, it is not hard to show that the

non-1 eigenvalues of ab are the non-1 eigenvalues ζa1m , . . . , ζacodim(a)m of a together with the non-

1 eigenvalues ζb1m , . . . , ζbcodim(b)m of b. If a, b 6= 1, we have a contradiction to p-connectedness of

ab, as p divides a1 + · · ·+ acodim(a) by virtue of a being in G(m, p, n). �

2.3.6. Lemma. Let g in G(m, p, n) be p-connected and not a diagonal reflection. Then there

exists a reflection s ≤⊥ g if and only if g has codimension two and non-1 eigenvalues ζcm and

ζ−cm for some c.

Proof. If g has codimension two and non-1 eigenvalues ζcm and ζ−cm , then g factors into

two reflections with codimensions adding. The factorization in dimension two illustrates the

general case: ζcm

ζ−cm

=

ζcm

ζ−cm

1

1

.

Conversely, if g is comparable with a reflection, then there must be a reflection s with

codim(gs) < codim(g). If s is a diagonal type reflection, then we get a contradiction to

p-connectedness of g. If s is a transposition type reflection, then, since g is diagonal, we use

Lemma 2.3.3 (2) to see that g must have nonzero cycle-sums ck and cl such that ck + cl ≡ 0

(mod m). By p-connectedness, ck and cl must be the only nonzero cycle-sums of g, and

hence the only non-1 eigenvalues of g are ζckm and ζclm = ζ−ckm . �

Since every element of G(m, p, n) must be above some atom, we now have the collection

of codimension atoms:

2.3.7. Proposition. The codimension atoms for G(m, p, n) are the reflections together with

the p-connected elements except for those with codimension two and determinant one.

It is known that length and codimension coincide for Coxeter groups and the family

G(m, 1, n), which, incidentally, includes the rank one groups G(m, p, 1) = G(mp, 1, 1). For

17

the remaining groups in the family G(m, p, n), we give explicit examples of codimension

atoms with reflection length exceeding codimension.

2.3.8. Corollary. The reflection length and codimension functions do not coincide in the

following groups:

• G(m, p, n) with 1 < p < m and n ≥ 2

• G(m,m, n) with m ≥ 3 and n ≥ 3.

Proof. Let Ik be the k × k identity matrix, and let M2 and M3 be the matrices

M2 =

ζm

ζp−1

m

and M3 =

ζm

ζ−2

m

ζm

.

In G(m, p, n) with 1 < p < m and n ≥ 2, the direct sum matrix M2 ⊕ In−2 has reflection

length three and codimension two. In G(m,m, n) with m ≥ 3 and n ≥ 3, the direct sum

matrix M3 ⊕ In−3 has reflection length four and codimension three. �

2.3.9. Remarks.

• A 1-connected element must be a diagonal reflection, so the set of codimension

atoms in G(m, 1, n) is simply the set of reflections. By Lemma 2.2.5, this recovers

the result that length and codimension coincide in G(m, 1, n).

• Shi [26] gives a formula for reflection length in G(m, p, n) in terms of a maximum

over certain partitions of cycle-sums. He also uses existence of a certain partition

of the cycle-sums as a necessary and sufficient condition for an element to have

reflection length equal to codimension.

2.4. Rank Two Exceptional Reflection Groups

The complex reflection groups G4−G22 act irreducibly on V ∼= C2. Each has at least one

conjugacy class of elements for which length and codimension differ. In all except for G8 and

G12, an argument comparing the order of the reflections with the order of the center of the

group demonstrates the existence of a central element with length greater than codimension.

18

2.4.1. Lemma. Let G be an irreducible complex reflection group acting on V ∼= C2. If

z ∈ G is central and G does not contain any reflections with the same order as z, then

`(z) > codim(z).

Proof. Since G acts irreducibly on V ∼= C2, each central element z 6= 1 is represented by a

scalar matrix of codimension two. Note that if z = st is a product of two reflections, then s

and t are actually commuting reflections. Then, working with s, t, and z simultaneously in

diagonal form, it is easy to deduce that the reflections s and t must have the same order as

z. Thus if G does not contain any reflections of the same order as z, we have `(z) > 2. �

Note that if g is a group element with codim(g) = 2, then `(g) = codim(g) if and only if

g can be expressed as a product of two reflections. Thus, we describe the codimension atoms

for a rank two reflection group:

2.4.2. Lemma. The codimension atoms in a rank two complex reflection group are the re-

flections together with the elements g such that `(g) > codim(g).

2.4.3. Proposition. Reflection length and codimension do not coincide in the rank two

exceptional complex reflection groups G4 −G22.

Proof. Inspection of Tables I, II, and III in Shephard-Todd [21] and an application of

Lemma 2.4.1 shows that each rank two group Gi with i 6= 8 or 12 has a central element z

such that `(z) > codim(z).

The group G8 can be generated by the order four reflections

r1 =

i

1

and r2 =1

2

1 + i 1 + i

−1− i 1 + i

.

The element

g = r1(r1r22r−11 )r2 =

1

2

−1 + i 1− i

−1− i −1− i

19

has length three and codimension two. Note that if g were the product of two reflections,

then gs would be a reflection for some reflection s in G8. However, computation shows

codim(gs) = 2 for all reflections s in G8.

For G12, let S and T be the generators given in Shephard-Todd [21]. Although S is a

reflection, the element T has codimension two. We express T as the product of two reflections

(each a conjugate of S):

T = (STST−1S−1)(T−1ST−1STS−1T ).

The element ST has length three and codimension two. (We verify the length by noting that

all reflections in G12 have determinant −1, so ST also has determinant −1 and must have

odd length.)

�

2.4.4. Remarks. Carter [2] proves length equals codimension in Weyl groups. Although

Carter’s proof applies equally well to any Coxeter group, we indicate two places where the

proof can break down for a general complex reflection group.

• Carter’s proof shows that in a real reflection group, if g has maximum codimension,

i.e., codim(g) = n, then codim(gs) < n for all reflections s in the group. This may

fail in a general complex reflection group, as illustrated by the element g in G8

given above in the proof of Proposition 2.4.3.

• Though G12 only has order two reflections, Carter’s proof fails for G12 because a

complex inner product is not symmetric.

2.5. Exceptional Reflection Groups G23 −G37

For the exceptional reflection groups, we work with the partial orders on the set CG of

conjugacy classes of G. With the aid of the software GAP [19], we compute reflection length,

atoms, and poset relations, appealing to character theory to speed up the computations.

Some of the exceptional reflection groups are Coxeter groups, for which reflection length

20

and codimension are known to agree. For the remaining groups, our computations show

reflection length and codimension do not coincide.

We first recall class algebra constants, which we use to aid our computations. Let X,

Y , and C be conjugacy classes of G, and let c be a fixed representative of C. The class

algebra constant ClassAlgConst(X, Y,C) counts the number of pairs (x, y) in X × Y such

that xy = c. These are the structure constants for the center of the group algebra and have

a formula in terms of the irreducible characters of G (details can be found in James and

Liebeck [14], for example).

Using class algebra constants, we can inductively find the elements of each reflection

length without having to multiply individual group elements. Let L(k) denote the set of

conjugacy classes whose elements have reflection length k. Suppose conjugacy class C is

not in L(0) ∪ · · · ∪ L(k) so that `(C) is at least k + 1. Then C is in L(k + 1) if and only

if ClassAlgConst(X, Y,C) is nonzero for some X in L(1) and Y in L(k). Since the class

algebra constants are nonnegative, we have `(C) = k + 1 if and only if

∑{ClassAlgConst(X, Y,C) : X ∈ L(1) and Y ∈ L(k)} 6= 0.

Using the same idea, we can easily compute all relations in the reflection length and

codimension posets on the set of conjugacy classes of G. For example, in the codimension

poset, we have

A ≤⊥ C ⇔∑X∈CG

codim(A)+codim(X)=codim(C)

ClassAlgConst(A,X,C) 6= 0.

In particular,

C is an atom in (CG,≤⊥) ⇔∑

X,Y ∈CG\{{1},C}codim(X)+codim(Y )=codim(C)

ClassAlgConst(X, Y,C) = 0.

Table 2.1 summarizes the data collected for the groups G23 −G37. (The Coxeter groups

are included for contrast.) The middle columns compare the number of conjugacy classes

of nonreflection atoms with the number of conjugacy classes C such that `(C) 6= codim(C).

The final columns compare maximum reflection length with the dimension n of the vector

21

group # conj classes # length 6=codim # nonref atoms dimV max ref length

23 10 0 0 3 3

24 12 2 2 3 4

25 24 3 1 3 4

26 48 9 5 3 4

27 34 12 12 3 5

28 25 0 0 4 4

29 37 10 4 4 6

30 34 0 0 4 4

31 59 27 5 4 6

32 102 27 6 4 6

33 40 12 6 5 7

34 169 78 14 6 10

35 25 0 0 6 6

36 60 0 0 7 7

37 112 0 0 8 8

Table 2.1. Atom Count in (CG,≤⊥) for Exceptional Reflection Groups G23 −G37

space on which the group acts. Note that in each case the maximum reflection length is at

most 2n− 1, usually less. In Appendix B, we record, for each exceptional reflection group, a

two-variable polynomial with the coefficient of the xlyd term indicating the number of group

elements with absolute reflection length l and codimension d.

2.6. When Does Reflection Length Equal Codimension?

Combining the existing results for Coxeter groups and G(m, 1, n) with our computations

for the remaining irreducible complex reflection groups, we complete the determination of

which reflection groups have length equal to codimension.

22

2.6.1. Theorem. Let G be an irreducible complex reflection group. The reflection length and

codimension functions coincide if and only if G is a Coxeter group or G = G(m, 1, n).

Proof. Carter’s proof [2] that reflection length coincides with codimension in Weyl groups

works just as well for Coxeter groups, and Shi [26] proves reflection length coincides with

codimension in the infinite family G(m, 1, n) (also see Shepler and Witherspoon [24] for a

more linear algebraic proof). For the converse, Corollary 2.3.8 gives counterexamples for

the remaining groups in the family G(m, p, n), while Proposition 2.4.3 and Table 2.1 show

reflection length and codimension do not coincide in the non-Coxeter exceptional complex

reflection groups. �

2.7. Applications to Hochschild Cohomology of Skew Group Algebras

The codimension poset has applications to Hochschild cohomology and deformation the-

ory of skew group algebras S(V )#G for a finite group G acting linearly on V . Defor-

mations of skew group algebras include graded Hecke algebras and symplectic reflection

algebras. Hochschild cohomology detects potential deformations. For a C-algebra A and an

A-bimodule M , the Hochschild cohomology of A with coefficients in M is the space

HH•(A,M) = Ext•A⊗Aop(A,M),

where we tensor over C. When M = A, we simply write HH•(A). We refer the reader

to Gerstenhaber and Schack [10] for more on algebraic deformation theory and Hochschild

cohomology.

In the setting of skew group algebras, Hochschild cohomology may be formulated in terms

of invariant theory. Stefan [27] finds cohomology of the skew group algebra S(V )#G as the

space of G-invariants in a larger cohomology ring:

HH•(S(V )#G) ∼= HH•(S(V ), S(V )#G)G,

and Farinati [7] and Ginzburg and Kaledin [11] describe the larger cohomology ring:

HH•(S(V ), S(V )#G) ∼=⊕g∈G

(S(V g)⊗

•−codim(g)∧(V g)∗ ⊗

codim(g)∧ ((V g)∗

)⊥ ⊗ Cg),

23

which we identify with a subspace of S(V ) ⊗∧• V ∗ ⊗ CG. Here, V g = {v ∈ V : gv = v}

denotes the fixed point space of g.

Shepler and Witherspoon [24] further show that the cohomology HH•(S(V ), S(V )#G)

is generated as an algebra under cup product by HH•(S(V )) together with derivation forms

corresponding to atoms in the codimension poset. More specifically, for each g in G, fix

a choice of volume form vol⊥g in the one-dimensional space∧codim(g)((V g)∗

)⊥. (If s is a

reflection, we may take vol⊥s in V ∗ to be a linear form defining the hyperplane about which

s reflects.) Then [24, Corollary 9.4] asserts that the cohomology ring HH•(S(V ), S(V )#G)

is generated by HH•(S(V )) ∼= S(V ) ⊗∧• V ∗ ⊗ 1G and the set of volume forms tagged by

codimension atoms:

{1⊗ vol⊥g ⊗g : g is an atom in the codimension poset for G}.

2.7.1. Example. Consider the group G = G(m, p, n) acting on V ∼= Cn by its standard

reflection representation. Let v1, . . . , vn denote the standard basis of V and v∗1, . . . , v∗n the

dual basis of V ∗. As in Section 2.3, let ζm = e2πi/m.

Proposition 2.3.7 describes the codimension atoms in G(m, p, n), and we can easily find

the corresponding volume forms vol⊥g . The cohomology HH•(S(V ), S(V )#G(m, p, n)) is thus

generated as a ring under cup product by HH•(S(V )) and the elements

• 1⊗ (v∗i − ζcmv∗j )⊗ s, where s is a reflection about the hyperplane v∗i − ζcmv∗j = 0, and

• 1⊗ v∗i1 ∧ · · · ∧ v∗icodim(g)

⊗ g, where g is p-connected and vi1 , . . . , vicodim(g)form a basis

of (V g)⊥.

(Note that we have included the elements 1 ⊗ (v∗i1 ∧ v∗i2

) ⊗ g with det(g) = 1, but these do

not arise from codimension atoms and are superfluous generators.)

Shepler and Witherspoon [24, Corollary 10.6] show that if G is a Coxeter group or

G = G(m, 1, n), then, in analogy with the Hochschild-Kostant-Rosenberg Theorem, the

cohomology HH•(S(V ), S(V )#G) is generated in cohomological degrees 0 and 1. We use

our comparison of the reflection length and codimension posets to show this analogue fails

for the other irreducible complex reflection groups.

24

We recall from [24, Section 8] the volume algebra Avol := SpanC{1⊗ vol⊥g ⊗g : g ∈ G},

isomorphic to a (generalized) twisted group algebra with multiplication

(1⊗ vol⊥g ⊗g) ^ (1⊗ vol⊥h ⊗h) = θ(g, h)(1⊗ vol⊥gh⊗gh)

for some cocycle θ : G×G→ C. The cocycle θ is generalized in that its values may include

zero; in fact, the twisting constant θ(g, h) is nonzero if and only if g ≤⊥ gh. Iterating the

product formula, we find

(1⊗ vol⊥g1 ⊗g1) ^ · · ·^ (1⊗ vol⊥gk ⊗gk) = λ(1⊗ vol⊥g1···gk ⊗g1 · · · gk),

where λ = θ(g1, g2)θ(g1g2, g3) · · · θ(g1 · · · gk−1, gk). The twisting constant λ is nonzero if and

only if g1 ≤⊥ g1g2 ≤⊥ · · · ≤⊥ g1 · · · gk. We make use of this fact in the proof of Lemma 2.7.2

below.

Once a choice of volume forms vol⊥g has been made, then given an element α in the

cohomology ring HH•(S(V ), S(V )#G), there exist unique elements αg in S(V g) ⊗∧•(V g)∗

such that

α =∑g∈G

αg ⊗ vol⊥g ⊗g.

Let the support of α be supp(α) = {g ∈ G : αg 6= 0}. For a set B ⊂ HH•(S(V ), S(V )#G),

let supp(B) =⋃β∈B supp(β). In the next lemma, we relate the support of a subring of

HH•(S(V ), S(V )#G) to the support of a set of generators for the subring.

2.7.2. Lemma. Let B be a subring of HH•(S(V ), S(V )#G), and let G(B) be a set of gener-

ators for B as a ring under cup product. If g is in supp(B), then there exist group elements

g1, . . . , gk in supp(G(B)) such that g1 ≤⊥ g1g2 ≤⊥ · · · ≤⊥ g1 · · · gk = g.

Proof. First consider the support of a finite cup product β1 ^ · · · ^ βk of generators

βi from G(B). Using the cup product formula [24, Equation (7.4)]2, we find that a typical

2Note that the factor dvg ∧ dvh in Equation (7.4) may not a priori be an element of∧•

(V gh)∗. To

interpret the equation correctly, we must apply to the wedge product dvg ∧ dvh the projection∧•

V ∗ →∧•(V gh)∗ induced by the orthogonal projection V ∗ → (V gh)∗. After the last iteration of the cup product

25

summand of β1 ^ · · ·^ βk has the form

ω ⊗ θ(g1, g2)θ(g1g2, g3) · · · θ(g1 · · · gk−1, gk) vol⊥g ⊗g,

where each gi is in supp(βi), g = g1 · · · gk, and ω is a (possibly zero) derivation form in

S(V g)⊗∧•(V g)∗. The scalar

θ(g1, g2)θ(g1g2, g3) · · · θ(g1 · · · gk−1, gk)

is a twisting constant from the volume algebra and, as noted above, is nonzero if and only if

g1 ≤⊥ g1g2 ≤⊥ · · · ≤⊥ g1 · · · gk. Thus

supp(β1 ^ · · ·^ βk) ⊆ {g1 · · · gk : gi ∈ supp(βi) and g1 ≤⊥ g1g2 ≤⊥ · · · ≤⊥ g1 · · · gk}.

Now note that for arbitrary elements α1, . . . , αk in B, we have

supp(α1 + · · ·+ αk) ⊆ supp(α1) ∪ · · · ∪ supp(αk).

This proves the lemma since every element of B is a sum of finite cup products of elements

of G(B). �

2.7.3. Corollary. The set of codimension atoms for G is contained in the support of every

generating set for HH•(S(V ), S(V )#G).

Proof. Let G be a set of generators for HH•(S(V ), S(V )#G). Applying Lemma 2.7.2, we

have that for each g 6= 1 in G there exist nonidentity group elements g1, . . . , gk in supp(G)

such that g1 ≤⊥ g1g2 ≤⊥ · · · ≤⊥ g1 · · · gk = g. In particular, g1 ≤⊥ g. If g is a codimension

atom, then since g1 6= 1 we must have g1 = g, and hence g lies in supp(G). �

2.7.4. Remark. The exterior products in the description of cohomology force a homogeneous

generator supported on a group element g to have cohomological degree at least codim(g)

(and no more than dimV = n). In light of Corollary 2.7.3, a set of homogeneous generators

for HH•(S(V ), S(V )#G) may well require elements of maximum cohomological degree n.

formula, we also apply the projections S(V ) → S(V )/I((V g)⊥) ∼= S(V g) to the polynomial parts to obtain

a representative in HH•(S(V ), S(V )#G).

26

For instance, in the group G(n, n, n) for n ≥ 3, the element g = diag(e2πi/n, . . . , e2πi/n) is a

codimension atom, and a homogeneous generator supported on g must have cohomological

degree codim(g) = n. Thus, using Corollary 2.7.3, we see that every set of homogeneous

generators for HH•(S(V ), S(V )#G(n, n, n)) includes an element of cohomological degree n.

2.7.5. Corollary. Let G be an irreducible complex reflection group. Then the cohomology

ring HH•(S(V ), S(V )#G) is generated in cohomological degrees 0 and 1 if and only if G is

a Coxeter group or a monomial reflection group G(m, 1, n).

Proof. By Corollary 2.7.3, the support in G of a set of generators for HH•(S(V ), S(V )#G)

must contain the set of codimension atoms. It follows that any set of generators contains

elements of cohomological degree at least as great as the codimensions of the atoms in

the codimension poset. If G is not a Coxeter group and not a monomial reflection group

G(m, 1, n), then there are nonreflection atoms in the codimension poset, so a generating set

for HH•(S(V ), S(V )#G) will necessarily include elements of cohomological degree greater

than one.

Conversely, Shepler and Witherspoon show in [24, Corollary 10.6] that if G is a Coxeter

group or a monomial reflection group G(m, 1, n), then HH•(S(V ), S(V )#G) can in fact be

generated in degrees 0 and 1. �

27

CHAPTER 3

TECHNIQUES FOR COMPUTING HOCHSCHILD COHOMOLOGY OF SKEW

GROUP ALGEBRAS

In this chapter, we outline various tools and observations that simplify Hochschild coho-

mology computations in the case of a skew group algebra.

3.1. Invariant Theory

In the setting of skew group algebras, Hochschild cohomology may be formulated in

terms of invariant theory. Let G be a finite group acting linearly on a complex vector space

V ∼= Cn. Without loss of generality, G preserves the inner product on V and acts by unitary

matrices. Stefan [27] finds cohomology of the skew group algebra S(V )#G as the space of

G-invariants in a larger cohomology ring:

HH•(S(V )#G) ∼= HH•(S(V ), S(V )#G)G,

and Farinati [7] and Ginzburg and Kaledin [11] describe the larger cohomology ring:

HH•(S(V ), S(V )#G) ∼=⊕g∈G

H•g⊗g,

where

H•g = S(V g)⊗•−codim(g)∧

(V g)∗ ⊗codim(g)∧

((V g)⊥)∗.

Here, V g = {v ∈ V : ~g(v) = v} denotes the fixed point space of g, and codim(g) := codimV g.

The group G acts diagonally on the tensor product, and the action on the tensor factor g is

by conjugation in the group. Note that the centralizer Z(g) of an element g in G preserves

the fixed point space V g and its orthogonal complement (V g)⊥, so the g-summand H•g⊗g is a

Z(g)-module. After switching the order of the operations “direct sum” and “take invariants”,

one can compute cohomology by finding invariants under centralizer subgroups and taking

28

a direct sum over a set of conjugacy class representatives (see Shepler-Witherspoon [23]). If

C (G) is any set of conjugacy class representatives of G, then

HH•(S(V )#G) ∼=

(⊕g∈G

H•g⊗g

)G

∼=⊕

g∈C (G)

(H•g⊗g

)Z(g).

We refer to the summand corresponding to the conjugacy class of g as the g-component

of Hochschild cohomology and denote it by HH•(g). Note that

HH•(g) ∼= (H•g)Z(g) =

S(V g)⊗•−codim(g)∧

(V g)∗ ⊗codim(g)∧

((V g)⊥)∗

Z(g)

,(1)

where the tensor factor g has been omitted since it carries a trivial Z(g)-action.

3.2. Actions of Centralizers on Fixed Point Spaces

Let χ be the character for a representation V ∼= Cn of a finite group G. For a group

element g in G, let χ ↓ Z(g) be the restriction of χ to the centralizer subgroup Z(g), and let

χ ↓ Z(g) = m1χ1 + · · ·+mrχr

be the unique decomposition of χ ↓ Z(g) into a combination of the irreducible characters

χ1, . . . , χr of Z(g). Because g is in the center of Z(g), Schur’s Lemma reveals that g acts

as a scalar multiple of the identity in each irreducible representation of Z(g). Thus, we can

pick out the constituents where g acts as the identity to get the character

χg =∑i

χi(g)=χi(1)

miχi

describing the action of Z(g) on the fixed point space V g, while the remaining constituents

give the character for the action of Z(g) on the orthogonal complement (V g)⊥.

3.3. Poincare Series

Poincare series (also called Hilbert or Molien series) are a tool for keeping track of the

dimensions of the pieces of a graded vector space. A graded vector space A =⊕∞

d=0Ad,

29

where each Ad is a finite dimensional complex vector space, has Poincare series

Px(A) =∞∑d=0

(dimCAd)xd.

3.3.1. Example. The Poincare series for the polynomial ring C[t] is

Px(C[t]) = 1 + x+ x2 + · · · = 1

1− x.

More generally, the Poincare series for a polynomial ring with generators td1 , . . . , tdr of ho-

mogeneous degrees d1, . . . , dr has Poincare series

Px(C[td1 , . . . , tdr ]) =1

(1− xd1) · · · (1− xdr).

Hochschild cohomology of a skew group algebra is bigraded by polynomial degree and

cohomological degree, so we seek a two-variable series with the coefficient of xiyj recording

the vector space dimension of the piece with polynomial degree i and cohomological degree j.

A generalization of Molien’s Theorem from invariant theory gives a generating function for

the two-variable Poincare series of the vector space (S(V )⊗∧• V ∗)G of invariant differential

forms (see, for example, Kane [15, Chapters 17 and 22]). Since the g-component of Hochschild

cohomology (Equation 1) may be expressed in terms of invariants in a tensor product, a slight

variation yields the Poincare series

Px,y(HH•(g)) =1

|Z(g)|∑h∈Z(g)

det−1(h⊥) det(1 + h∗y)

det(1− hx)ycodim(g),

where h⊥ is the action of h on (V g)⊥, h∗ in the numerator acts on (V g)∗, and h in the

denominator acts on V g. In turn, the Poincare series for S(V )#G is

Px,y (HH•(S(V )#G)) =∑

g∈C (G)

Px,y(HH•(g)).

In Section 5.3.1, we use Poincare series in an example to verify that we have found a complete

set of generators for the identity component of Hochschild cohomology.

30

3.4. Necessary Conditions for Nonzero Cohomology

For convenience, we record some necessary conditions in order to have HHk(g) 6= 0.

3.4.1. Observation. Let G be any finite group acting linearly on V ∼= Cn. If the cohomology

component HHk(g) is nonzero, then det(g) = 1 and codim(g) ≤ k ≤ n.

Proof. Note that g acts trivially on the tensor product S(V g)⊗∧k−codim(g)(V g)∗ and scales

the one-dimensional vector space∧codim(g)((V g)⊥)∗ by det−1(g). So necessarily det(g) = 1

in order to have any nonzero Z(g)-invariants in H•g. For the second criteria, note that the

exterior factor∧k−codim(g)(V g)∗ is zero unless 0 ≤ k − codim(g) ≤ n− codim(g). �

Based on the above observation, the following table illustrates, for n = 4, which compo-

nents of cohomology may be potentially nonzero.

codim(g) 0 1 2 3 4

HH0(g) * 0 0 0 0

HH1(g) * 0 0 0 0

HH2(g) * 0 * 0 0

HH3(g) * 0 * * 0

HH4(g) * 0 * * *

Note that elements of codimension one (reflections) never have determinant one, so based on

the codimension restrictions, HH2(g) can be nontrivial only for elements with codimension

zero or two.

3.5. Cohomology Component for an Element with an Abelian Centralizer

If the centralizer of g in G is abelian, then the g-component of Hochschild cohomology

can be described in terms of polynomials semi-invariant under various linear characters.

We first establish some notation. Let G be a finite group acting linearly on V ∼= Cn, and

suppose the centralizer of g in G is abelian. Then Z(g) acts diagonally on V with respect

to some basis, say v1, . . . , vn. We may assume the vectors are ordered so that ~g(vi) = vi

if and only if 1 ≤ i ≤ dimV g. Let k be a cohomological degree with codim(g) ≤ k ≤ n,

and let p = k − codim(g). (Note that 0 ≤ p ≤ dimV g.) Define an eigenvector basis of

31

∧k−codim(g)(V g)∗ ⊗∧codim(g)((V g)⊥)∗ as follows: For each tuple I = (i1, . . . , ip) in the set

Ik = {(i1, . . . , ip) : 1 ≤ i1 < · · · < ip ≤ dimV g},

define the element v∗I in∧k−codim(g)(V g)∗ ⊗

∧codim(g)((V g)⊥)∗ by

v∗I = v∗i1 ∧ · · · ∧ v∗ip ⊗ vol⊥g ,

where vol⊥g = 1 if dimV g = n and vol⊥g = v∗dimV g+1 ∧ · · · ∧ v∗n if dimV g < n. By con-

vention, Icodim(g) = {()} and v∗() = 1 ⊗ vol⊥g . Then {v∗I}I∈Ik is a Z(g)-eigenvector basis for∧k−codim(g)(V g)∗⊗∧codim(g)((V g)⊥)∗, and for each tuple I in Ik, there exists a linear character

χI : Z(g)→ C× such that

~h(v∗I ) = χI(h)v∗I

for every h in the centralizer Z(g).

3.5.1. Observation. Let G be a finite group acting linearly on a vector space V ∼= Cn, and

suppose the centralizer of g in G is abelian. Then, using the notation above, the g-component

of Hochschild cohomology of the skew group algebra S(V )#G is given by

HHk(g) ∼=

⊕I∈Ik

S(V g)χI ⊗ v∗I if det(g) = 1 and codim(g) ≤ k ≤ n

0 otherwise,

where S(V g)χI = {f ∈ S(V g) : ~h(f) = χI(h)f for all h in Z(g)}.

3.5.2. Observation. Let G be a finite group with a cyclic center generated by an element

z. Suppose G acts irreducibly on a vector space V ∼= Cn so that z scales vectors in V by

some primitive rth root of unity ζr. If the centralizer of g in G is Z(g) = 〈g, z〉, then the

g-component of Hochschild cohomology of S(V )#G is given by

HHk(g) ∼=

⊕I∈Ik,

d≡k (mod r)

Sd(V g)⊗ v∗I if det(g) = 1 and codim(g) ≤ k ≤ n

0 otherwise,

where Sd(V g) is the subspace of S(V g) spanned by the degree d monomials.

32

Proof. Suppose det(g) = 1 and codim(g) ≤ k ≤ n (otherwise HHk(g) = 0 by Observa-

tion 3.4.1). The g-component of Hochschild cohomology is

HHk(g) ∼=

S(V g)⊗k−codim(g)∧

(V g)∗ ⊗codim(g)∧

((V g)⊥)∗

Z(g)

∼=

⊕I∈Ikd≥0

Sd(V g)⊗ Cv∗I

Z(g)

.

Since Z(g) preserves the summands Sd(V g)⊗ Cv∗I , it suffices to consider the action of Z(g)

on an element f ⊗ v∗I , where f is a homogeneous polynomial in S(V g). Since z scales vectors

in V by ζr and vectors in V ∗ by ζ−1r , we have

~g(f ⊗ v∗I ) = f ⊗ det−1(g)v∗I = f ⊗ v∗I and ~z(f ⊗ v∗I ) = ζdeg(f)r f ⊗ ζ−kr v∗I .

Thus, f ⊗ v∗I is Z(g)-invariant if and only if deg(f)− k ≡ 0 mod r. �

3.6. Cohomology Component for an Element with a Trivial Fixed Point Space

If g is a group element with maximum codimension codim(g) = n, then we can describe

the g-component of Hochschild cohomology explicitly just by knowing the determinants of

the elements in the centralizer of g.

3.6.1. Lemma. Let G be a finite group acting linearly on a vector space V ∼= Cn with basis

v1, . . . , vn. Suppose g is a group element with maximum codimension codim(g) = n. Then

HHk(g) = 0 for k 6= n, and

HHn(g) ∼=

Cv∗1 ∧ · · · ∧ v∗n if det(h) = 1 for all h in Z(g)

0 otherwise.

Proof. Let g be an element of G with codim(g) = n. Then HHk(g) = 0 when k 6= n by

Observation 3.4.1. Turning to the case k = n, we have

HHn(g) ∼=

S(V g)⊗n−codim(g)∧

(V g)∗ ⊗codim(g)∧

((V g)⊥)∗

Z(g)

=

(C⊗ C⊗

n∧V ∗

)Z(g)

.

33

The volume form vol⊥g = v∗1 ∧ · · · ∧ v∗n is a basis for∧n V ∗, and

~h(1⊗ 1⊗ vol⊥g ) = det−1(h)(1⊗ 1⊗ vol⊥g )

for every h in Z(g). Thus 1⊗ 1⊗ vol⊥g is Z(g)-invariant if and only if det(h) = 1 for all h in

Z(g). �

3.7. Galois Conjugate Representations

If the action of a finite group G on a finite-dimensional vector space V is represented

by matrices with entries in the field Q(ζ) for some root of unity ζ, then applying a field

automorphism γ to the entries of the matrices gives another representation Vγ of G, not nec-

essarily equivalent to the representation V . However, since the Poincare series for Hochschild

cohomology of a skew group algebra is given in terms of characteristic polynomials and de-

terminants of matrices (see Section 3.3), we have

Px,y(HH•(S(Vγ)#G)) = γ (Px,y(HH•(S(V )#G))) .

Since the coefficients in the Poincare series are integers, applying the field automorphism γ

leaves the series unchanged, and Px,y(HH•(S(Vγ)#G)) = Px,y(HH•(S(V )#G)). In particular,

if the g-component of one Hochschild cohomology ring is zero, then the g-component of the

other must also be zero. This is a convenient observation when deciding which skew group

algebras to investigate.

3.8. Representations Differing by a Group Automorphism

It is possible for inequivalent representations to give rise to isomorphic skew group alge-

bras. For example, if two representations of a group differ by a group automorphism, then

the skew group algebras are isomorphic, and so is the resulting cohomology.

Let G be a finite group acting on V ∼= Cn, say by representation ρ : G → GL(V ). For

any group automorphism φ of G, composing with ρ gives another representation ρ ◦ φ of G.

We write Vφ to indicate when the action of G on V is the composition ρ ◦ φ. An isomorphism

between the skew group algebras S(V )#G and S(Vφ)#G is given by v 7→ v and g 7→ φ−1(g),

34

and an explicit conversion between their Hochschild cohomology rings is accomplished by

merely changing group element tags.

35

CHAPTER 4

RANK TWO EXCEPTIONAL REFLECTION GROUPS

Since the centralizers of the rank two complex reflection groups G4 − G22 are easy to

describe, we can get a fairly good idea of what the Hochschild cohomology looks like. In this

chapter, we describe the centralizers of elements in each rank two complex reflection group

and then give methods for using GAP (Groups, Algorithms, and Programming) to compute

Hochschild cohomology.

4.1. Centralizer Subgroups of Reflection Groups G4 −G22

In any group G, the smallest possible centralizer of an element g is the subgroup generated

by g and the center of G, while the largest possible centralizer is the whole group. We say

an element g has a medium centralizer if

〈g, Z(G)〉 ( Z(g) ( G.

In the tetrahedral and icosahedral complex reflection groups, there are no elements with

medium centralizers:

4.1.1. Lemma. Let G be a tetrahedral complex reflection group (one of G4 − G7 in the

Shephard-Todd table) or an icosahedral group (one of G16−G22 in the Shephard-Todd table).

Let z be a generator of the (cyclic) center of G. If g is a noncentral element of G, then

Z(g) = 〈g, z〉.

Proof. This is easily verified by using GAP to compare Size(Centralizer(G,g)) with

Size(G) and Size(Closure(Centre(G),g)). �

In an octahedral reflection group, there are elements with medium centralizers. Some

elements with medium centralizers are squares, and upon computing the centralizer of a

particular square element h2 to be Z(h2) = 〈h, z〉, we notice that every element with a

36

medium centralizer is conjugate to an element of the h2Z coset, leading to the following

description of all medium centralizers:

4.1.2. Lemma. Let G be an octahedral complex reflection group (one of G8 − G15 in the

Shephard-Todd table). Let z be a generator of the (cyclic) center Z of G. If g in G is an

element with a medium centralizer, i.e., 〈g, z〉 ( Z(g) ( G, then there exists an element h

in G such that g is in the coset h2Z and Z(g) = Z(h) = 〈h, z〉.

Proof. Let g be an element with a medium centralizer. Suppose h is an element with

centralizer Z(h) = 〈h, z〉 and |Z(h)| = |Z(g)|. If g is in the coset h2Z, then Z(g) ⊇ Z(h),

and, by comparing sizes, Z(g) = Z(h). In fact, it suffices to know that g is conjugate to an

element of the coset h2Z. For, if g = xh2zmx−1 for some power m and some x in G, then

the hypotheses of the first argument hold with h replaced by xhx−1.

Using the software GAP [19] with the CHEVIE [9] package for complex reflection groups,

one can verify the hypothesis: for each g with a medium centralizer, there exists an element

h such that Z(h) = 〈h, z〉, |Z(h)| = |Z(g)|, and g is conjugate to an element of the coset

h2Z. �

4.2. Cohomology for Irreducible Representations

If G is any of the groups G4 − G22 acting irreducibly on V ∼= Cn, then, since the

noncentral elements in G have small abelian centralizers, we can compute all nonidentity

components of Hochschild cohomology of S(V )#G using the techniques in Chapter 3. This

can be automated using the software GAP [19], where we have access to character tables,

conjugacy classes, centralizers, and eigenvalues. The CHEVIE [9] package is used to load

complex reflection groups.

Let G be any of the groups G4 −G22. Given an irreducible character of G, a conjugacy

class representative g, and a cohomological degree k, we automate the following steps to

determine HHk(g):

(i) Compute det(g) and codim(g) with the help of the Eigenvalues command.

37

(ii) Apply Observation 3.4.1. If det(g) = 1 and codim(g) ≤ k ≤ n, then continue to

the next step. Otherwise, record HHk(g) = 0.

(iii) If codim(g) < n, then continue to the next step. Otherwise, record HHk(g) using

Lemma 3.6.1.

(iv) Compare Size(Centralizer(G,g)) with Size(Closure(Centre(G),g)) and with

Size(G) to determine the centralizer type.

(v) If Z(g) = 〈g, z〉, apply Observation 3.5.2.

(vi) If g has a medium centralizer, then find an element z that generates the center

of G, and find the element h guaranteed by Lemma 4.1.2. Then use the idea in

Section 3.2 to compute the linear characters needed to apply Observation 3.5.1.

(vii) If Z(g) = G, then, since the representation of G is irreducible and we have already

determined codim(g) < n, the element g must act as the identity. Now,

HH•(g) ∼= HH•(1) ∼=

(S(V )⊗

•∧V ∗

)G

,

which our code does not describe explicitly. (See Section 5.3.1 for an example

using standard invariant theory techniques to determine the identity component

by hand.)

4.2.1. Example. To illustrate, a summary of the Hochschild cohomology of S(V )#G where

G = G6 acts irreducibly on V ∼= Cn (with n > 1) is included in Appendix C. The group

G6 has fourteen irreducible representations with characters labeled X.1, . . . ,X.14. We omit

the cohomology tables for the six one-dimensional representations since all components of

Hochschild 2-cohomology are zero, implying that there are no nontrivial deformations of the

corresponding skew group algebras (see Gerstenhaber and Schack [10]).

In each of the cohomology tables, an entry ∗ denotes a cohomology component

HHk(g) ∼=

(S(V )⊗

k∧V ∗

)G

,

isomorphic to the identity component of degree k cohomology.

38

The zeros in the cohomology tables for all of the two-dimensional representations may be

deduced from Observation 3.4.1 and Lemma 3.6.1. Only the identity component is nontrivial.

Representations X.7, X.8, X.11, and X.12 are reflection representations.

The zeros in the cohomology tables for X.13 and X.14 follow from the determinant and

codimension restrictions given in Observation 3.4.1. An entry of the form kmr denotes a

cohomology component

HHk(g) ∼=⊕I∈Ik,

d≡k mod r

Sd(V g)⊗ Cv∗I

given by Observation 3.5.2. The congruences are unsimplified to emphasize that they can

be read off from the cohomological degree and the action of the center.

Hochschild 2-cohomology contains all possible first multiplication maps of an associative

deformation of an algebra. Looking at the cohomology tables in the G6 example, a few

representations stand out: X.7, X.8, X.11, and X.12 (reflection representations) have a

scarcity of nontrivial Hochschild 2-cohomology, while X.13 (a “rotation representation”)

has an abundance of nontrivial Hochschild 2-cohomology. The analogous representations

are present for each of the groups G4 − G22. In Section 4.3, we compute the Hochschild

cohomology for the reflection representation of each of the rank two exceptional reflection

groups. In Chapter 5, we define a “rotation representation” for each of the groups G4−G22

and compute the corresponding cohomology in the case of the tetrahedral groups G4 −G7.

4.3. Cohomology for the Reflection Representation

In this section, we describe the Hochschild cohomology for the skew group algebra

S(V )#G, where G is any of the groups G4 − G22 acting on V ∼= C2 by its reflection repre-

sentation.

39

4.3.1. Proposition. Let Gi with 4 ≤ i ≤ 22 be a rank two exceptional complex reflection

group acting on V ∼= C2 by its standard irreducible reflection representation. Then

HH•(S(V )#Gi) ∼=

HH•(1) if i 6= 4, 12

HH•(1)⊕ HH•(g4) if i = 4

HH•(1)⊕ HH•(g6)⊕ HH•(g26) if i = 12,

where

• g4 is a representative of the conjugacy class of order four elements in G4, and

• g6 is a representative of the conjugacy class of order six elements in G12.

Descriptions of the cohomology components are indicated in the proof.

Proof. Let g be an element of one of the rank two reflection groups Gi.

If codim(g) = 0, then g = 1, and HHk(1) ∼=(S(V )⊗

∧k V ∗)G

, which is well-studied for

the reflection representation (see Orlik and Terao [17]).

If codim(g) = 1, then HHk(g) = 0 for all k by Observation 3.4.1.

If codim(g) = 2, then g has a trivial fixed point space. By Lemma 3.6.1, HHk(g) = 0 for

k 6= 2, and

HH2(g) ∼=

Cv∗1 ∧ v∗2 if det(h) = 1 for all h ∈ Z(g)

0 otherwise.

To determine whether HH2(g) 6= 0 for some elements, one considers the groups Gi individ-

ually in an argument that parallels Section 2C of Ram and Shepler [18]. In fact, Ram and

Shepler were investigating the polynomial degree zero part of Hochschild 2-cohomology, and

the formula above shows there is no cohomology with higher polynomial degree, so their

computation of HH20(g) completely describes HH2(g) when codim(g) = 2. �

40

CHAPTER 5

COHOMOLOGY FOR LIFTS OF QUOTIENT GROUP REPRESENTATIONS

In this chapter, we consider the deformation theory of skew group algebras S(V )#G and

S(V )#G/K, where the action of the group G on the vector space V is lifted from an action

of a quotient group G/K on V . After observing some general simplifications, we apply

them to the specific example of tetrahedral complex reflection groups acting on V ∼= C3 by

a nonfaithful “rotation representation” lifted from a representation of the quotient group

G/Z ∼= Alt4.

5.1. The Lazy Mathematician’s Dream Formula

Before investigating the tetrahedral example, we consider the general setup. Let K

be a normal subgroup of a finite group G and π : G � G/K the canonical quotient ho-

momorphism. Suppose G/K acts on a vector space V ∼= Cn, say by a representation

ρ : G/K → GL(V ). Then the composition Gπ→ G/K

ρ→ GL(V ) is a representation of

G, and we can compare the deformation theory of the skew group algebras S(V )#G and

S(V )#G/K. Keeping in mind that a conjugacy class in the quotient group G/K lifts to a

union of conjugacy classes in G, the lazy mathematician’s dream formula would be

HH•(S(V )#G) ∼=⊕

gK∈C (G/K)

HH•(gK)⊕mg ,

where C (G/K) is a set of conjugacy class representatives for the quotient group G/K, and

the multiplicity mg is the number of conjugacy classes of G that intersect the coset gK

nontrivially. This formula, however, breaks down in the following two ways:

(i) The conjugacy class of g in G and the conjugacy class of gK in G/K need not give

isomorphic contributions to HH•(S(V )#G) and HH•(S(V )#G/K), respectively.

(ii) Conjugacy classes of elements in the same coset of K need not give isomorphic

contributions to the cohomology of S(V )#G.

41

In this section we highlight sufficient conditions for (i) and (ii) to hold, giving us techniques

for efficient computations in the tetrahedral example in Section 5.4.

Recall from Section 3.1 that the g-component of Hochschild cohomology of the skew

group algebra S(V )#G is HH•(g) ∼=(H•g)Z(g)

, where

H•g = S(V g)⊗•−codim(g)∧

(V g)∗ ⊗codim(g)∧

((V g)⊥)∗.

Since G acts on V by ρ ◦ π, and G/K acts on V by ρ, we have

V g = {v ∈ V : ((ρ ◦ π)(g))(v) = v} = {v ∈ V : ρ(π(g))(v) = v} = V π(g)

and H•g = H•π(g). The g-component of HH•(S(V )#G) is

HH•(g) ∼= (H•g)ZG(g) = (H•π(g))

π(ZG(g)),

while the π(g)-component of HH•(S(V )#G/K) is

HH•(π(g)) ∼= (H•π(g))Zπ(G)(π(g)).

When the centralizers π(ZG(g)) and Zπ(G)(π(g)) are equal, we have HH•(g) ∼= HH•(π(g)):

5.1.1. Proposition. Let K be a normal subgroup of the finite group G. If the image of

the centralizer ZG(g) under the canonical quotient homomorphism G → G/K is all of the

centralizer ZG/K(gK), then the contribution HH•(g) to the cohomology HH•(S(V )#G) is

isomorphic to the contribution HH•(gK) to the cohomology HH•(S(V )#G/K).

Unfortunately, π(ZG(g)) is often a proper subgroup of Zπ(G)(π(g)). In this case the space

of π(ZG(g))-invariants is typically larger than the space of Zπ(G)(π(g))-invariants (fewer

group elements, easier to be invariant), and we cannot directly describe the contribution

HH•(g) in terms of the contribution HH•(π(g)).

5.1.2. Example. The order 48 reflection group G6 has an order four center Z, and the

quotient group G6/Z is isomorphic to the alternating group Alt4. The diagram here shows

how the fourteen conjugacy classes of G6 project down onto the four conjugacy classes of

Alt4.

42

The conjugacy classes of G6 are denoted by 1a, 4a, etc. (The number in a label indicates the

order of each element in the conjugacy class.) Each column represents a coset of the center,

and the labels in that column record the conjugacy classes of the elements in the coset.

Suppose G6 acts on a vector space V by a representation with kernel Z, the center of the

group. A priori,

HH•(S(V )#G6) ∼=⊕

g∈C (G6)

HH•(g),

a sum over the fourteen conjugacy classes of G6. However, elements of the same coset of the

kernel Z act the same on V , so they have the same fixed point space. Furthermore, elements

in the same coset of the center always have the same centralizer. Thus, HH•(g) ∼= HH•(g′)

if g and g′ are in the same coset of the center. Now, we can shorten our work by writing

HH•(S(V )#G6) ∼= HH•(1a)⊕4 ⊕ HH•(4a)⊕2 ⊕ HH•(12c)⊕4 ⊕ HH•(12b)⊕4,

a sum of four components with multiplicities.

5.1.3. Proposition. Let G be a finite group acting linearly on a vectorspace V ∼= Cn, and

suppose the kernel K = {g ∈ G : ~g(v) = v for all v in V} of the action is a subset of the

center of the group G. Let C (G/K) be a set of conjugacy class representatives of the quotient

group G/K. Then the Hochschild cohomology of the skew group algebra S(V )#G is

HH•(S(V )#G) ∼=⊕

gK∈C (G/K)

HH•(g)⊕mg ,

where the multiplicity mg =|ZG(g)||ZG/K(gK)|

is the number of conjugacy classes of G that inter-

sect the coset gK nontrivially.

43

Proof. Suppose g and g′ are in the same coset of the kernel K. Then they act the same on

V , so their fixed point spaces are the same. Furthermore, since K is a subset of the center,

g and g′ have the same centralizer. Thus, HH•(g) ∼= HH•(g′). It follows that

HH•(S(V )#G) ∼=⊕

gK∈C (G/K)

HH•(g)⊕mg ,

where the multiplicity mg is the number of conjugacy classes that intersect the coset gK

nontrivially.

We compute the multiplicities in terms of centralizer sizes. Let g be an element of G.

Recall from group theory that the conjugacy class ClG/K(gK) = {gK, g1K, . . . , grK} lifts to

a union of conjugacy classes in G. Furthermore, the elements of the conjugacy class ClG(g)

of g in G distribute equally across the cosets gK, g1K, . . . , grK, so

|ClG(g) ∩ gK| = |ClG(g)||ClG/K(gK)|

.

If g and g′ are in the same coset of the central kernel K, then |ClG(g)∩gK| = |ClG(g′)∩gK|,

as g and g′ have the same centralizer (and hence the same conjugacy class sizes). Now the

number of conjugacy classes intersecting the coset gK nontrivially is

mg =|K|

|ClG(g) ∩ gK|= |K| ·

|ClG/K(gK)||ClG(g)|

=|K||G|

|K||ZG/K(gK)|· |ZG(g)||G|

=|ZG(g)||ZG/K(gK)|

,

as required. �

The following example illustrates that for an arbitrary nonfaithful action, elements of the

same coset of the kernel of the action may give nonisomorphic contributions to cohomology.

5.1.4. Example. Consider the dihedral group D4 generated by an order four rotation r and

a reflection f . Define a nonfaithful action of D4 on V ∼= C2 by letting elements of the

rotation subgroup {1, r, r2, r3} act as the identity and elements of the coset {f, rf, r2f, r3f}

act by the matrix diag(−1, 1) with respect to a basis v1, v2 of V . Then

HH•(r) ∼=

(S(V )⊗

•∧V ∗

)〈r〉= S(V )⊗

•∧V ∗,

44

but

HH•(1) ∼=

(S(V )⊗

•∧V ∗

)D4

( S(V )⊗•∧V ∗.

The last inclusion is proper since, for example, HH2(1) ∼= {f ⊗ v∗1 ∧ v∗2 : f is in v1C[v21, v2]}.

5.2. The Rotation Representation

Complex reflection groupsG4−G22 in the Shephard-Todd classification share the property

that their central quotient is the rotational symmetry group of a Platonic solid. Write G/Z

for the quotient of a group by its center. For groups G4−G7, the quotient G/Z is isomorphic

to Alt4, the rotational symmetry group of the tetrahedron. For groups G8−G15, the quotient

G/Z is isomorphic to Sym4, the rotational symmetry group of the octahedron. For groups

G16 − G22, the quotient G/Z is isomorphic to Alt5, the rotational symmetry group of the

icosahedron.

Let G be any of the groups G4 −G22, and write Z for the center of G. Let V ∼= C3, and

let ρ : G/Z → GL(V ) be the complexification of the natural rotation action of G/Z on R3.

We call the composition Gπ→ G/Z

ρ→ GL(V ) the rotation representation of G.

The remainder of this chapter is devoted to cohomology for the skew group algebras of

the alternating group on four letters and the tetrahedral reflection groups G4 − G7, each

acting on V ∼= C3 by the rotation representation. Although we only include full details

for the tetrahedral groups, a similar analysis could be carried out for the octahedral and

icosahedral groups acting by the rotation representation.

5.3. Cohomology for the Alternating Group on Four Letters

Let Alt4 denote the alternating group on four letters. The symmetric group Sym4 acts on

the vector space V ∼= C4 by permuting the standard basis vectors e1, e2, e3, e4. The subspace

L = SpanC{e1+e2+e3+e4} is an invariant line, and an orthonormal basis for the orthogonal

complement L⊥ is

v1 =1

2(e1 − e2 + e3 − e4)

v2 =1

2(e1 − e2 − e3 + e4)

45

v3 =1

2(e1 + e2 − e3 − e4).

The group Alt4 acts irreducibly on V := L⊥, and with respect to the basis v1, v2, v3 we have

generating matrices

(3 4 2) 7→

1

1 0

0 1

and (3 2 1) 7→

−1

−1 0

0 1

.

The alternating group Alt4 has four conjugacy classes, so there are four components of

Hochschild cohomology to compute:

HH•(S(V )# Alt4) ∼= HH•(1)⊕ HH•((12)(34))⊕ HH•(123)⊕ HH•(321).

Subsection 5.3.1 is devoted to the identity component; Subsection 5.3.2 describes the com-

ponent for (12)(34); and Subsection 5.3.3 describes the components for the three-cycles.

5.3.1. Identity Component

The cohomology contribution from the identity is HH•(1) ∼= (S(V )⊗∧• V ∗)Alt4 . One

standard approach to computing Alt4-invariants is to observe that if W is a finite dimensional

Sym4-module, then

WAlt4 = W Sym4 ⊕W Sym4sgn ,

where WSym4sgn = {w ∈ W : ~g(w) = sgn(g)w for all g in Sym4}. (This decomposition follows

from the fact that Alt4 is an index two subgroup of Sym4.) We choose instead a direct (and

perhaps longer) approach, as it nicely highlights a wide range of techniques from invariant

theory and is instructive to the nonexpert.

Let G = Alt4. We first compute the Poincare series for HH•(1), one for each cohomolog-

ical degree:

Px((S(V )⊗0∧V ∗)G) =

1 + x6

(1− x2)(1− x3)(1− x4)

Px((S(V )⊗1∧V ∗)G) =

x+ x2 + 2x3 + x4 + x5

(1− x2)(1− x3)(1− x4)

46

Px((S(V )⊗2∧V ∗)G) =

x+ x2 + 2x3 + x4 + x5

(1− x2)(1− x3)(1− x4)

Px((S(V )⊗3∧V ∗)G) =

1 + x6

(1− x2)(1− x3)(1− x4)

The denominator for each Poincare series is (1 − x2)(1 − x3)(1 − x4), suggesting that

(S(V )⊗∧• V ∗)G is a finitely generated free C[f2, f3, f4]-module for some algebraically inde-

pendent invariant polynomials f2, f3, and f4 of degrees 2, 3, and 4, respectively. Since the

group action is given by monomial matrices, it is easy to guess invariant polynomials with

these degrees:

f2 = v21 + v22 + v23

f3 = v1v2v3

f4 = v41 + v42 + v43.

The Jacobian determinant

det

(∂fi∂vj

)= det

2v1 2v2 2v3

v2v3 v1v3 v1v2

4v31 4v32 4v33

.= v41(v22 − v23) + v42(v23 − v21) + v43(v21 − v22)

is a nonzero polynomial, so f2, f3, and f4 are algebraically independent.

The numerators of the Poincare series predict the polynomial degrees of the secondary

generators of (S(V ) ⊗∧k V ∗)G. (The identity component of cohomology will be the free

C[f2, f3, f4]-span of the secondary generators.) In cohomological degrees zero and three, the

numerator (1 + x6) predicts generators of polynomial degrees 0 and 6, and in cohomological

degrees one and two, the numerator (x+x2+2x3+x4+x5) predicts generators of polynomial

degrees 1, 2, 3, 3, 4, 5.

Before constructing secondary generators, we establish some notation. Define ordered

bases Bk of∧k V ∗ as follows:

B0 = {1}

47

B1 = {v∗1, v∗2, v∗3}

B2 = {v∗2 ∧ v∗3, v∗3 ∧ v∗1, v∗1 ∧ v∗2}

B3 = {v∗1 ∧ v∗2 ∧ v∗3}.

Note that∧0 V ∗ and

∧3 V ∗ both carry the trivial representation of Alt4, while∧1 V ∗ and∧2 V ∗ are isomorphic to V as Alt4-modules. In fact, an easy check reveals that the matrices

for the action of Alt4 on V with respect to the basis {v1, v2, v3} also serve as matrices for

the action of Alt4 on the spaces∧1 V ∗ and

∧2 V ∗ with respect to the bases B1 and B2,

respectively.

For 0 ≤ k ≤ 3, let nk =(3k

)denote the size of the basis Bk = {b1, . . . , bnk} of

∧k V ∗.

Given an nk-tuple of polynomials pi, let (p1, . . . , pnk)Bk denote the element∑pi ⊗ bi of

S(V )⊗∧k V ∗.

The following standard key facts motivate the construction of Alt4-invariant forms:

(i) Wedging two G-invariant forms produces another G-invariant form.

(ii) Suppose W and W ∗ are dual irreducible representations of a group G. If the

sets B = {w1, . . . , wn} and B∗ = {w∗1, . . . , w∗n} are bases of W and W ∗ such that

[g]B∗ = [g]B, then (W ⊗W ∗)G is spanned by∑n

i=1wi ⊗ w∗i .

When a representation is self-dual, the gradient of an invariant polynomial determines

an invariant derivation. Using the invariant polynomials f2, f3, f4 from above, construct

invariant derivations by scaling ( ∂fi∂v1, ∂fi∂v2, ∂fi∂v3

):

α1,1 = (v1, v2, v3)B1

α2,1 = (v2v3, v3v1, v1v2)B1

α3,1 = (v31, v32, v

33)B1 .

Wedge them together two at a time to get invariant 2-forms of polynomial degrees 3, 4, 5:

β3,2 := α1,1α2,1 = (v1(v22 − v23), v2(v

23 − v21), v3(v

21 − v22))B2

β4,2 := α3,1α1,1 = (v2v3(v22 − v23), v1v3(v

23 − v21), v1v2(v

21 − v22))B2

48

β5,2 := α2,1α3,1 = (v1(v42 − v43), v2(v

43 − v41), v3(v

41 − v42))B2 .

Wedge them together three at a time to get an invariant 3-form of polynomial degree 6:

f6,3 := α1,1α2,1α3,1 = (v41(v22 − v23) + v42(v23 − v21) + v43(v21 − v22))B3 .

(Up to a scalar, this is the Jacobian determinant from earlier.) The indices in the above

notation indicate the polynomial and cohomological degrees of the elements. For example,

β4,2 has polynomial degree 4 and cohomological degree 2.

So far, we have only half of the required generators. But, since there is an Alt4-module

isomorphism V ∼= V ∗ ∼=∧2 V ∗, we are able to use the polynomials from above and simply

change the bases to obtain the remaining generators. For i = 1, 2, 3, define the 2-form αi,2

by switching the basis on αi,1 from B1 to B2. So, for example, α1,2 = (v1, v2, v3)B2 . Similarly,

for i = 3, 4, 5, define the 1-form βi,1 by switching the basis on βi,2 from B2 to B1. Finally,

identify (S(V )⊗∧0 V ∗) with S(V ), and let f6 = f6,0 be the polynomial part of f6,3.

We claim all of the elements αi,k and βi,k are Alt4-invariant. To illustrate, consider α2,1

and α2,2. The set P = {v2v3, v3v1, v1v2} of polynomial parts is a basis for an Alt4-stable

subspace of S(V ). It is easily verified that

[g]P = [g]B1 = [g]B2 = [g]B1 = [g]B2 ,

so, by the second key fact, both α2,1 and α2,2 are Alt4-invariant. Similarly, one can show that

all of the forms αi,k and βi,k are Alt4-invariant. Note that f6 is an Alt4-invariant polynomial.

5.3.1.1. Claim. Let Alt4 act on V ∼= C3 by the rotation representation. Let R = C[f2, f3, f4].

Then the identity component of HH•(S(V )# Alt4) is given by

HH0(1) ∼= R⊕Rf6

HH1(1) ∼= Rα1,1 ⊕Rα2,1 ⊕Rα3,1 ⊕Rβ3,1 ⊕Rβ4,1 ⊕Rα5,1

HH2(1) ∼= Rα1,2 ⊕Rα2,2 ⊕Rα3,2 ⊕Rβ3,2 ⊕Rβ4,2 ⊕Rα5,2

HH3(1) ∼= Rv∗1 ∧ v∗2 ∧ v∗3 ⊕Rf6,3.

49

In each cohomological degree k, we have already verified that the proposed secondary

generators are Alt4-invariant, so the R-span of the generators on the right-hand side is a

subset of HHk(1). By showing the sums on the right are direct, i.e. the generators are R-

linearly independent, we will have the same Poincare series for each side, and since one side

is already a subset of the other, we will have equality.

The next step is to verify that in each cohomological degree the proposed secondary

generators are C[f2, f3, f4]-linearly independent. The trick for doing this is to extend the

action of the alternating group Alt4 on V to an action of the symmetric group Sym4 on V .

Recalling that the basis vectors vi were initially defined in terms of basis vectors ei naturally

permuted by the symmetric group, we can generate Sym4 with the additional matrix

(1 2) 7→

0 −1

−1 0

1

.

Let σ = (1 2). Note that the polynomials f2, f3, f4 are invariant under σ (and hence

Sym4), but f6 is not: ~σ(f6) = −f6. We first show 1 and f6 are C[f2, f3, f4]-linearly indepen-

dent. Suppose p + qf6 = 0 for some polynomials p, q in C[f2, f3, f4]. Applying σ to both

sides of the equation yields p− qf6 = 0. Now qf6 = −qf6 forcing q = 0 and, in turn, p = 0.

Thus 1 and f6 are C[f2, f3, f4]-linearly independent, and R + Rf6 ⊆ HH0(1) is a direct sum

with the same Poincare series as HH0(1), so HH0(1) ∼= R⊕Rf6.

The same idea shows that the αi,1’s and βi,1’s are R-linearly independent. First note that

the αi,1’s are invariant under σ, but the βi,1’s are only semi-invariant:

~σ(αi,1) = αi,1 and ~σ(βi,1) = −βi,1.

If

p1α1,1 + p2α2,1 + p3α3,1 + q3β3,1 + q4β4,1 + q5β5,1 = 0

50

for some invariant polynomials pi and qi in C[f2, f3, f4], then applying σ to the equation

gives us also that

p1α1,1 + p2α2,1 + p3α3,1 − q3β3,1 − q4β4,1 − q5β5,1 = 0.

Then q3β3,1 + q4β4,1 + q5β5,1 = 0, and in turn p1α1,1 + p2α2,1 + p3α3,1 = 0. Expanding out

what this means gives us two matrix equations:

(p1 p2 p3

)v1 v2 v3

v2v3 v3v1 v1v2

v31 v32 v33

=(

0 0 0)

and

(q3 q4 q5

)v1(v

22 − v23) v2(v

23 − v21) v3(v

21 − v22)

v2v3(v22 − v23) v1v3(v

23 − v21) v1v2(v

21 − v22)

v1(v42 − v43) v2(v

43 − v41) v3(v

41 − v42)

=(

0 0 0).

But the determinants of the 3 × 3 matrices are nonzero polynomials, so we must have

p1 = p2 = p3 = 0 and q3 = q4 = q5 = 0. Then

Rα1,1 +Rα2,1 +Rα3,1 +Rβ3,1 +Rβ4,1 +Rβ5,1 ⊆ HH1(1)

is actually a direct sum with the same Poincare series as HH1(1), so

HH1(1) ∼= Rα1,1 ⊕Rα2,1 ⊕Rα3,1 ⊕Rβ3,1 ⊕Rβ4,1 ⊕Rα5,1.

Analogous arguments establish Claim 5.3.1.1 for cohomological degrees two and three. Note

that there is a role reversal in degree two: the βi,2’s are invariant under σ, while the αi,2’s

are only semi-invariant.

5.3.2. Double-flip Component

In this subsection, we compute the Hochschild cohomology component for the conjugacy

class of double-flips. Let g = (13)(24). The centralizer of g in Alt4 is the copy of the Klein

51

four group generated by the matrices

(13)(24) 7→

1

−1

−1

and (14)(23) 7→

−1

1

−1

.

By Observation 3.4.1, HHk((13)(24)) = 0 for k 6= 2, 3. The two-form vol⊥g = v∗2 ∧ v∗3 is a

basis for∧2((V g)⊥)∗. In the following table, we record the information needed to calculate

the invariants describing the Hochschild cohomology contribution from the conjugacy class

of g:

generators of Z(g) (13)(24) (14)(23)

action on V g = Cv1 1 −1

action on∧2((V g)⊥)∗ = C vol⊥g 1 −1

From this information, we find that

HH2((13)(24)) ∼=(S(V g)⊗ C⊗

2∧((V g)⊥

)∗ )Z((13)(24))=

⊕d2≡1

(Sd(V g)⊗ C⊗

2∧((V g)⊥

)∗ ).

∼= v1 C[v21]⊗ C⊗ C vol⊥g

and

HH3((13)(24)) ∼=(S(V g)⊗ (V g)∗ ⊗

2∧((V g)⊥

)∗ )Z((13)(24))=

⊕d2≡0

(Sd(V g)⊗ (V g)∗ ⊗

2∧((V g)⊥

)∗ ).

∼= C[v21]⊗ Cv∗1 ⊗ C vol⊥g

Thus, HH2(g) is the C[v21]-span of v1⊗1⊗vol⊥g , and HH3(g) is the C[v21]-span of 1⊗v∗1⊗vol⊥g .

5.3.3. Three-cycle Components

In this subsection, we compute the Hochschild cohomology components for the conjugacy

classes of g = (243) and g−1 = (342). The choice of representative does not matter; we simply

52

chose these because of the simplicity of the basis for V g:

V g = SpanC{v1 + v2 + v3} and (V g)⊥ = SpanC{v1 + ωv2 + ω2v3, v1 + ω2v2 + ωv3},

where ω = e2πi/3. The two-form vol⊥g = v∗1 ∧ v∗2 + v∗2 ∧ v∗3 + v∗3 ∧ v∗1 is a basis for the one

dimensional space∧2((V g)⊥)∗.

Because g has determinant one and generates its own centralizer, we have

HHk(g) ∼=

S(V g)⊗k−codim(g)∧

(V g)∗ ⊗codim(g)∧

((V g)⊥)∗

Z(g)

= S(V g)⊗k−codim(g)∧

(V g)∗ ⊗codim(g)∧

((V g)⊥)∗.

By Observation 3.4.1, HHk(g) = 0 for k 6= 2, 3. For k = 2, 3, we describe HHk(g) with

respect to an explicit basis. HH2(g) is isomorphic to the C[v1 + v2 + v3]-span of 1⊗ 1⊗ vol⊥g ,

and HH3(g) is isomorphic to the C[v1 + v2 + v3]-span of 1⊗ (v∗1 + v∗2 + v∗3)⊗ vol⊥g .

5.4. Cohomology for Tetrahedral Groups Acting by the Rotation Representation

We now apply the techniques of Section 5.1 and, whenever possible, use components

from the cohomology of S(V )# Alt4 (see Section 5.3) to determine Hochschild cohomology

for any tetrahedral group G4 −G7 acting on V ∼= C3 by the rotation representation defined

in Section 5.2.

We first use the software GAP [19] to record the centralizer sizes necessary to apply

Propositions 5.1.1 and 5.1.3. Let Z denote the center of G, and let π : G→ G/Z ∼= Alt4 be

the canonical quotient homomorphism.

Centralizer sizes for tetrahedral groups G4 −G7

conjugacy class of π(g) 1 (13)(24) (243) (342)

size of ZG(g) 12|Z| 2|Z| 3|Z| 3|Z|

size of π(ZG(g)) 12 2 3 3

size of ZG/Z(π(g)) 12 4 3 3

The sizes of the centers of the groups G4, G5, G6, and G7 are 2, 6, 4, and 12, respectively.

53

5.4.1. Proposition. Let G be any of the tetrahedral groups G4 − G7 acting on V ∼= C3 by

the rotation representation, and let Z denote the center of G. Then

HH•(S(V )#G) ∼= HH•(g1)⊕|Z| ⊕ HH•(g2)

⊕|Z|/2 ⊕ HH•(g3)⊕|Z| ⊕ HH•(g−13 )⊕|Z|

∼= HH•(1)⊕|Z| ⊕ HH•(g2)⊕|Z|/2 ⊕ HH•(243)⊕|Z| ⊕ HH•(342)⊕|Z|,

where g1 = 1, and g2 and g3 in G are chosen so that π(g2) = (13)(24) and π(g3) = (243).

Explicit descriptions of the components are indicated in the proof.

Proof. Note that the kernel of the rotation representation for G is the center of the group.

Thus Proposition 5.1.3 applies, and we read the multiplicities mg =|ZG(g)||ZG/Z(gZ)|

off of the

table of centralizer sizes. Secondly, when |π(ZG(g))| = |ZAlt4(π(g))|, then we may apply

Proposition 5.1.1 and use components from the cohomology of S(V )# Alt4.

The component HH•(1) is given in Subsection 5.3.1, and the components HH•(243) and

HH•(342) are given in Subsection 5.3.3. Since g2 has determinant one in the rotation repre-

sentation and generates its own centralizer in G, we have

HHk(g2) ∼=

S(V g2)⊗k−codim(g2)∧

(V g2)∗ ⊗codim(g2)∧

((V g2)⊥)∗

ZG(g2)

= S(V g2)⊗k−codim(g2)∧

(V g2)∗ ⊗codim(g2)∧

((V g2)⊥)∗.

If we choose the conjugacy class representative g2 in G such that π(g2) = (13)(24), then,

using the matrices from Section 5.3.1, V g2 = Cv1. By Observation 3.4.1, HHk(g2) = 0 for

k 6= 2, 3. For degrees two and three, we have that HH2(g2) is the C[v1]-span of 1⊗ 1⊗ vol⊥g2 ,

and HH3(g2) is the C[v1]-span of 1⊗ v∗1 ⊗ vol⊥g2 , where vol⊥g2 = v∗2 ∧ v∗3. �

Shepler and Witherspoon have shown in [23, Theorem 8.7] that the parameter space

of graded Hecke algebras for a skew group algebra is isomorphic (as a C-vector space) to

the polynomial degree zero part of Hochschild 2-cohomology. We apply the theorem to the

rotation representation of the tetrahedral groups:

54

5.4.2. Corollary. Let G be any of the tetrahedral groups G4−G7 acting on V ∼= C3 by the

rotation representation, and let Z denote the center of G. The parameter space of graded

Hecke algebras for S(V )#G is 5|Z|2

-dimensional (as a vector space over C).

5.4.3. Remark. While the parameter space of graded Hecke algebras for S(V )# Alt4, where

Alt4 acts on V ∼= C3 by the rotation representation, is only two-dimensional, there are four

dimensions (as a C-vector space) worth of cohomology in polynomial degree one, cohomo-

logical degree two. It would be interesting to investigate these in the context of the Drinfeld

orbifold algebras discussed in Shepler-Witherspoon [22].

55

CHAPTER 6

REGULAR REPRESENTATION

In this chapter, we classify and give an explicit description of all graded Hecke algebras

obtained as deformations of the skew group algebra S(V )#G where the finite group G acts

on V ∼= C|G| by the regular representation. With respect to the basis {vg : g ∈ G} of C|G|,

the left regular action of a group element g on a vector vx is denoted by ~g(vx) = vgx.

Shepler and Witherspoon showed in [23, Theorem 8.7] that the parameter space of graded

Hecke algebras for a skew group algebra is isomorphic (as a vector space over C) to the

polynomial degree zero part of Hochschild 2-cohomology, which we denote by HH20(S(V )#G).

Recall from Section 3.1 that the g-component of Hochschild cohomology of a skew group

algebra S(V )#G is the space of centralizer invariants

HH•(g) ∼=

S(V g)⊗•−codim(g)∧

(V g)∗ ⊗codim(g)∧

((V g)⊥)∗

Z(g)

.

The formula simplifies considerably when restricting to polynomial degree zero and cohomo-

logical degree two. For convenience, we record the simplifications in an observation, omitting

any one-dimensional tensor factors on which the centralizer is guaranteed to act trivially.

6.0.4. Observation. Let G be a finite group acting linearly on a vector space V ∼= Cn. Then

the g-component of HH20(S(V )#G) is

HH20(g) ∼=

(∧2 V ∗

)Z(g)if codim(g) = 0(∧2((V g)⊥)∗

)Z(g)if codim(g) = 2

0 otherwise.

In the formula above, the cases codim(g) 6= 0, 2 follow from Observation 3.4.1.

56

6.1. G-invariant Two-forms

Consider the identity component in polynomial degree zero and cohomological degree

two:

HH20(1) ∼=

(2∧V ∗

)G

.

We first use inner products of characters to compute the vector space dimension of

HH20(1). Let χreg be the character for the action of G on V (and also V ∗ since the regular

representation is self-dual); let χAlt2 be the character for the action of G on∧2 V ∗; and let ι

be the trivial character. Then, by character theory (see [20], for example), the dimension of

HH20(1) is

dimC

(2∧V ∗

)G

=⟨χAlt2 , ι

⟩=

1

|G|∑g∈G

χAlt2 (g)ι(g)

=1

|G|∑g∈G

χAlt2 (g)

=1

|G|∑g∈G

χ2reg(g)− χreg(g2)

2.

Recalling that χreg(1) = |G| and χreg(g) = 0 for g 6= 1, the summands are zero except when

g2 = 1. Thus

dimC

(2∧V ∗

)G

=1

|G|∑g∈Gg2=1

χ2reg(g)− χreg(g2)

2

=1

2|G|

∑g∈Gg2=1

χ2reg(g)−

∑g∈Gg2=1

χreg(g2)

=

1

2|G|(|G|2 − |G| ·#{g ∈ G : g2 = 1}

)=|G| −#{g ∈ G : g2 = 1}

2.

57

6.1.1. Lemma. Let G be a finite group acting on V ∼= C|G| by the regular representation.

Then the vector space dimension of the polynomial degree zero part of the identity component

of HH2(S(V )#G) is

dimC HH20(1) =

|G| −#{g ∈ G : g2 = 1}2

.

The next task is to explicitly describe a basis for(∧2 V ∗

)G. A standard technique for

producing G-invariants is averaging over the group.

6.1.2. Lemma. Let G act on V ∼= C|G| by the left regular representation. For each g in G,

define a G-invariant two-form αg in(∧2 V ∗

)Gby

αg =∑x∈G

~x(v∗1 ∧ v∗g) =∑x∈G

v∗x ∧ v∗xg.

Partition G\{g : g2 = 1} into two-element sets {g, g−1}, and let B be a subset of G containing

exactly one representative from each block of the partition. Then

(i) αg−1 = −αg,

(ii) αg = 0 if and only if g2 = 1, and

(iii) the set {αg : g ∈ B} is a basis of(∧2 V ∗

)G.

Proof. (i) Note that for fixed g in G, {x : x ∈ G} = {xg−1 : x ∈ G}, so

αg =∑x∈G

~x(v∗1 ∧ v∗g)

=∑x∈G

# »

xg−1(v∗1 ∧ v∗g)

=∑x∈G

~x(v∗g−1 ∧ v∗1)

=∑x∈G

~x(−v∗1 ∧ v∗g−1)

= −αg−1 .

(ii) If αg = 0, then there exist group elements x and y such that yg = x and y = xg. But

then y = xg = (yg)g, and g2 = 1. Conversely, if g2 = 1, then g = g−1 and αg = 0 by part

(i).

58

(iii) By (ii), αg 6= 0 for all g in B. To show {αg : g ∈ B} is a linearly independent set, it

suffices to show that for distinct group elements g and h in B, the sets {±v∗x ∧ v∗xg : x ∈ G}

and {±v∗y ∧ v∗yh : y ∈ G} are disjoint.

Suppose the intersection {±v∗x ∧ v∗xg : x ∈ G} ∩ {±v∗y ∧ v∗yh : y ∈ G} is nonempty. Then

there exist elements x and y such that x = y and xg = yh, or there exist elements x and y

such that x = yh and xg = y. The first case leads to g = h, while the second case leads to

g = h−1, both contradictions if g and h are distinct elements of B. Now, since the size of

the linearly independent set {αg : g ∈ B} equals dimC(∧2 V ∗)G, we have our basis. �

6.2. Polynomial Degree Zero Hochschild 2-cohomology

In this section, we compute the components HH20(g) for the nonidentity group elements g.

Combined with the previous section, this yields a complete description of HH20(S(V )#G) and

allows us to apply Theorem 8.7 of Shepler and Witherspoon [23] to determine the dimension

of the parameter space of graded Hecke algebras for S(V )#G when G acts by the regular

representation.

For most groups, HH20(g) is zero for the nonidentity elements:

6.2.1. Proposition. Let G act on V ∼= C|G| by the regular representation. If |G| > 4, then

HH20(S(V )#G) ∼= HH2

0(1),

and the parameter space of graded Hecke algebras for S(V )#G has dimension

|G| − {g ∈ G : g2 = 1}2

as a vector space over C. A basis of HH20(1) is given in Lemma 6.1.2.

Proof. We show that codim(g) > 2 for all nonidentity group elements g. Then, by Obser-

vation 3.4.1, HH2(g) = 0 for all g 6= 1.

Suppose g 6= 1, and let |g| denote the order g. Then

codim(g) = dimV − dimV g = |G| − |G||g|

= |G|(

1− 1

|g|

)≥ |G|

2> 2,

where we have used |g| > 1 and |G| > 4 to conclude the inequalities. �

59

There are four remaining groups to consider: the Klein four group and the cyclic groups

of orders two, three, and four. We consider the cyclic group of order two and the Klein four

group as special cases of an arbitrary finite direct product of Z/2Z.

6.2.2. Proposition. Let G ∼= Z/2Z× · · · × Z/2Z︸ ︷︷ ︸m

act on V ∼= C2m by the regular represen-

tation. Then

HH20(S(V )#G) = 0,

and S(V )#G has no nontrivial graded Hecke algebras.

Proof. Since every element of G squares to the identity, we have HH20(1) = 0. To show the

nonidentity components are also zero, we consider the three cases m = 1, m = 2, and m ≥ 3

separately.

If m = 1, then write G = {1, a}. The nonidentity element a has determinant −1 in the

regular representation, so HH20(a) is zero by Observation 3.4.1.

If m = 2, then write G = {1, a, b, ab}. Each element is in its own conjugacy class, so

there are three nonidentity components to consider. We show details for the a-component,

and the others are computed similarly. Note that

V a = SpanC{v1 + va, vb + vab} and (V a)⊥ = SpanC{v1 − va, vb − vab}.

Thus, codim(a) = 2, and, by Observation 6.0.4,

HH20(a) ∼=

(2∧

((V a)⊥)∗

)Z(a)

.

The two-form vol⊥a = (v∗1 − v∗a)∧ (v∗b − v∗ab) is a basis for∧2((V a)⊥)∗. However, b centralizes

a and does not fix vol⊥a , so we have HH20(a) = 0. Similarly, one can argue HH2

0(b) = 0 and

HH20(ab) = 0.

If m ≥ 3, then |G| > 4 and, by Proposition 6.2.1, HH20(g) = 0 for all g 6= 1.

We have now shown HH20(S(V )#G) = 0, so there are no nontrivial graded Hecke algebras

for S(V )#G.

�

60

The cyclic groups of orders three and four have nonzero contribution from some noniden-

tity elements.

6.2.3. Proposition. Let G ∼= Z/3Z act on V ∼= C3 by the regular representation. Write

G = {1, g, g2}. Then

HH20(S(V )#G) ∼= HH2

0(1)⊕ HH20(g)⊕ HH2

0(g2),

where HH20(1) ∼= HH2

0(g) ∼= HH20(g

2) ∼= Cαg with αg = v∗1 ∧ v∗g + v∗g ∧ v∗g2 + v∗g2 ∧ v∗1. The

parameter space of graded Hecke algebras for S(V )#G is 3-dimensional.

Proof. Since each element of G is in its own conjugacy class, there are two nonidentity

components to compute. However, since g and g−1 share the same fixed point space and

centralizer, HH•(g) ∼= HH•(g−1). Note that

V g = SpanC{v1 + vg + vg2} and (V g)⊥ = SpanC{v1 + ωvg + ω2vg2 , v1 + ω2vg + ωvg2},

where ω = e2πi/3. Thus, codim(g) = 2, and, by Observation 6.0.4,

HH20(g) ∼=

(2∧

((V g)⊥)∗

)Z(g)

.

The two-form αg = v∗1 ∧v∗g +v∗g ∧v∗g2 +v∗g2 ∧v∗1 is a basis of∧2((V g)⊥)∗ and is invariant under

Z(g) = G. Hence, HH20(g) ∼= HH2

0(g−1) ∼= Cαg. By Lemma 6.1.2, αg also spans HH2

0(1). �

6.2.4. Proposition. Let G ∼= Z/4Z act on V ∼= C4 by the regular representation, and let g

be a generator of G. Then

HH20(S(V )#G) ∼= HH2

0(1)⊕ HH20(g

2),

where HH20(1) ∼= HH2

0(g2) ∼= Cαg with αg = v∗1 ∧ v∗g + v∗g ∧ v∗g2 + v∗g2 ∧ v∗g3 + v∗g3 ∧ v∗1. The

parameter space of graded Hecke algebras for S(V )#G is 2-dimensional.

Proof. Each element of G is in its own conjugacy class, so there are three nonidentity

components to compute. Since g and g−1 are order four, their eigenvalues in the regular rep-

resentation are 1, i,−1,−i. Hence codim(g) = codim(g−1) = 3, and HH20(g) = HH2

0(g−1) = 0

61

by Observation 6.0.4. However, the eigenvalues of g2 are 1, 1,−1,−1, so codim(g) = 2 and

HH20(g

2) ∼=

(2∧

((V g2)⊥)∗

)Z(g2)

.

Note that

V g2 = SpanC{v1 + vg2 , vg + vg3} and (V g2)⊥ = SpanC{v1 − vg2 , vg − vg3}.

The two-form αg = v∗1 ∧ v∗g + v∗g ∧ v∗g2 + v∗g2 ∧ v∗g3 + v∗g3 ∧ v∗1 is a basis of∧2((V g2)⊥)∗ and

is invariant under Z(g) = G. Thus, HH20(g

2) ∼= Cαg. By Lemma 6.1.2, αg also spans

HH20(1). �

6.3. From Cohomology to Graded Hecke Algebras

In this section, we use Theorem 11.4 and Proposition 11.5 of Shepler and Witherspoon [25]

to convert each element of HH20(S(V )#G) into a skew-symmetric bilinear form

κ : V × V → CG

that defines a graded Hecke algebra

Hκ = T (V )#G/Iκ,

where Iκ is the two-sided ideal Iκ = 〈v ⊗ w − w ⊗ v − κ(v, w) : v, w ∈ V 〉. The work of

Shepler and Witherspoon shows that this conversion yields all graded Hecke algebras that

are deformations of the skew group algebra S(V )#G.

Before proceeding, we establish some notational conventions. First, note that for a given

map κ : V × V → CG, there exist unique maps κg : V × V → C such that κ decomposes as

the sum

κ =∑g∈G

κgg.

Secondly, recall that if W is any finite dimensional vector space, then the space∧2W ∗

identifies with the space of skew-symmetric bilinear forms from W × W into C. After

choosing a basis of W , say w1, . . . , wn, a bilinear form α : W ×W → C can be described by

a matrix [α] with (i, j)-entry equal to α(wi, wj).

62

6.3.1. Lemma. Let G be a finite group. Fix an ordering of the basis {vg : g ∈ G} of

V ∼= C|G|, and let [g]R denote the matrix of the group element g acting on V by the right

regular representation: ~g(vx) = vxg−1. Then the matrix of the two-form αg =∑

x∈G v∗x ∧ v∗xg

in∧2 V ∗ is

[αg] = [g]R − [g]ᵀR.

Proof. If g2 = 1, then [αg] = 0 by part (i) of Lemma 6.1.2. On the other hand, [g]R is a

permutation matrix and g = g−1, so [g]R − [g]ᵀR = [g]R − [g−1]R = 0, as required.

If g2 6= 1, then

αg(vx, vy) =

1 if y = xg

−1 if x = yg

0 otherwise.

The (vx, vy)-entry of [g]R is 1 if vyg−1 = vx and 0 otherwise. Transposing, the (vx, vy)-entry

of [g]ᵀR is 1 if vxg−1 = vy and 0 otherwise. It follows that the (vx, vy)-entry of [g]R − [g]ᵀR is

precisely αg(vx, vy). �

6.3.2. Example. Let the quaternion group Q8 = {±1,±i,±j,±k} act on V ∼= C8 by the

left regular representation, and let v1, v−1, vi, v−i, vj, v−j, vk, v−k be an ordered basis of V . In

Proposition 6.2.1, we showed

HH20(S(V )#Q8) ∼= HH2

0(1) ∼= SpanC{αi, αj, αk},

so, by Proposition 11.5 of [25], the maps κ : V × V → CQ8 that define a graded Hecke

algebra Hκ are precisely the maps

κ = aαi + bαj + cαk,

63

where a, b, c are any complex scalars. The matrix of κ is

[κ] = [κ1] =

0 0 a −a b −b c −c

0 0 −a a −b b −c c

−a a 0 0 −c c b −b

a −a 0 0 c −c −b b

−b b c −c 0 0 −a a

b −b −c c 0 0 a −a

−c c −b b a −a 0 0

c −c b −b −a a 0 0

.

The parameter space of graded Hecke algebras is 3-dimensional.

6.3.3. Example. Let the cyclic group G = {1, g, g2} act on V ∼= C3 by the left regular

representation, and let v1, vg, vg2 be an ordered basis of V ∼= C3. In Proposition 6.2.3, we

showed that

HH20(1) ∼= HH2

0(g) ∼= HH20(g

2) ∼= Cαg,

so, by Proposition 11.5 of [25], the maps κ : V ×V → CG that define a graded Hecke algebra

Hκ are precisely the maps

κ = aαg + bαgg + cαgg2,

where a, b, c are any complex scalars. The matrix of αg is

[αg] = [g]R − [g]ᵀR =

0 1 −1

−1 0 1

1 −1 0

,

so each element a + bg + cg2 in the group algebra determines a graded Hecke algebra with

commutator relations

[v1, vg] = [vg, vg2 ] = [vg2 , v1] = a+ bg + cg2.

The parameter space of graded Hecke algebras is 3-dimensional.

64

6.3.4. Example. Let the cyclic group G = {1, g, g2, g3} act on V ∼= C4 by the left regular

representation, and let v1, vg, vg2 , vg3 be an ordered basis of V . In Proposition 6.2.4, we

showed

HH20(1) ∼= HH2

0(g2) ∼= Cαg and HH2

0(g) ∼= HH20(g

3) = 0,

so, by Proposition 11.5 of [25], the maps κ : V ×V → CG that define a graded Hecke algebra

Hκ are precisely the maps

κ = aαg + bαgg2,

where a and b are any complex scalars. The matrix of αg is

[αg] = [g]R − [g]ᵀR =

0 1 0 −1

−1 0 1 0

0 −1 0 1

1 0 −1 0

,

so each element a + bg2 in the group algebra determines a graded Hecke algebra with com-

mutator relations

[v1, vg] = [vg, vg2 ] = [vg2 , vg3 ] = [vg3 , v1] = a+ bg2

and

[v1, vg2 ] = [vg, vg3 ] = 0.

The parameter space of graded Hecke algebras is 2-dimensional.

65

66

APPENDIX A

REFLECTION LENGTH AND CODIMENSION CODE

To compute reflection length and atoms in the codimension poset (see Chapter 2), we

wrote the following code that runs using the software GAP [19]. The code requires the

CHEVIE package [9] to load complex reflection groups and their character tables. Note,

however, that our functions LengthPartition and LengthClassFunction do not require

the group to be a reflection group and can be used to compute length with respect to any

set of generators closed under conjugation.

##########################################################################

# lengthcodim.gap

##########################################################################

# WANT TO COMPUTE REFLECTION LENGTH?

# 1. Start a GAP3 session and load chevie

# gap> LoadPackage("chevie");

# 2. Read in this file lengthcodim.gap from wherever you saved it, e.g.

# gap> Read("mygapfolder/lengthcodim.gap");

# 3. Define your complex reflection group G4-G37, e.g.

# gap> G:=ComplexReflectionGroup(4);

# 4. Get the character table for your group

# gap> t:=CharTable(G);

# 5. If your group is a Coxeter group (23,28,30,35,36,37), then you

# need to attach the group to the character table manually:

# gap> t.group:=G;

# 6. Determine which character corresponds to the reflection rep

# gap> refrep:=GetRefRep(t);

# 7. Determine which conjugacy classes contain reflections

# gap> refclasses:=RefClasses(t,refrep);

# 8. Find absolute reflection length!

# gap> LengthClassFunction(t,refclasses);

#

# NOTE: Steps 6-8 may be done all at once:

# gap> L:=LengthClassFunction(t,RefClasses(t,GetRefRep(t)));

67

##########################################################################

# FUNCTIONS IN THIS FILE

# This file contains functions for comparing reflection length and

# codimension posets for complex reflection groups.

#-------------------------------------------------------------------------

# ClassNmbrToName:=function(charTable,classNmbr)

# ClassToNmbr:=function(charTable,class)

# ClassToName:=function(charTable,class)

# ConjClassMultTable:=function(charTable,classList1,classList2)

# IrrCodim:=function(charTable,charNmbr,classNmbr)

# CodimClassFunction:=function(charTable,charNmbr)

# CodimPartition:=function(charTable,charNmbr)

# IsMinimalInP:=function(charTable,charNmbr,classk,P)

# MinimalClassFunction:=function(charTable,charNmbr)

# LengthPartition:=function(charTable, genClassNums)

# LengthClassFunction:=function(charTable,genClassNums)

# RefClasses:=function(charTable,charNmbr)

# RefClassFunction:=function(charTable,charNmbr)

# GetRefRep:=function(charTable)

#-------------------------------------------------------------------------

RequirePackage("chevie");

##########################################################################

# CLASS NMBR TO NAME

# This function converts the CLASSNMBR of a conjugacy class (position in

# list of conjugacy classes attached to CHARTABLE into its name used in

# the display of CHARTABLE

#-------------------------------------------------------------------------

ClassNmbrToName:=function(charTable,classNmbr)

ClassNamesCharTable(charTable);

return charTable.classnames[classNmbr];

end;

##########################################################################

# CLASS TO NMBR

# This function returns the position of a conjugacy CLASS in the list of

# conjugacy classes attached to CHARTABLE

#-------------------------------------------------------------------------

ClassToNmbr:=function(charTable,class)

return Position(ConjugacyClasses(charTable.group),class);

68

end;

##########################################################################

# CLASS TO NAME

# This function returns the class name for a conjugacy CLASS in the list

# of conjugacy classes attached to CHARTABLE

#-------------------------------------------------------------------------

ClassToName:=function(charTable,class)

return ClassNmbrToName(charTable,ClassToNmbr(charTable,class));

end;

##########################################################################

# CONJUGACY CLASS MULTIPLICATION TABLE

# Input: CHARTABLE - the character table of a group G

# CLASSLIST1 - a list of positions of conjugacy classes

# in charTable.classes

# CLASSLIST2 - a second list of positions of conjugacy classes in

# charTable.classes

#

# Output: this function prints out the multiplication table for

# multiplying conjugacy classes in the first list with the

# conjugacy classes in the second list

#-------------------------------------------------------------------------

ConjClassMultTable:=function(charTable,classList1,classList2)

local c1, c2, result, k;

for c1 in classList1 do

for c2 in classList2 do

# pick out the class numbers k having nonzero

# ClassAlgConst(c1,c2,k)

result:=Filtered([1..Length(charTable.irreducibles)],

k->ClassMultCoeffCharTable(charTable,c1,c2,k)>0);

Print(ClassNmbrToName(charTable,c1),"*",

ClassNmbrToName(charTable,c2),":",

List(result,k->ClassNmbrToName(charTable,k)),"\n");

od;

Print("\n");

69

od;

end;

##########################################################################

# IRR CODIM

# Input: CHARTABLE - the character table of a group G

# CHARNMBR - position of an irreducible character in

# charTable.irreducibles

# CLASSNMBR - position of a conjugacy class in charTable.classes

#

# Output: codimension of subspace fixed by an elt in the conjugacy class

# given by CLASSNMBR in representation with specified character

#

# WARNING: THIS FUNCTION ASSUMES THE IDENTITY IS IN THE FIRST COLUMN

# OF THE CHARTABLE

#-------------------------------------------------------------------------

IrrCodim:=function(charTable,charNmbr,classNmbr)

local evals;

evals:=Eigenvalues(charTable,

charTable.irreducibles[charNmbr],classNmbr);

return Eigenvalues(charTable,charTable.irreducibles[charNmbr],1)[1]

-evals[Length(evals)];

end;

##########################################################################

# CODIM CLASS FUNCTION

# Input: CHARTABLE - character table for a group

# CHARNMBR - position of an irreducible character in

# charTable.irreducibles

#

# Output: a class function with g -> codim(V^g),

# where V^g=subspace fixed by g

#-------------------------------------------------------------------------

CodimClassFunction:=function(charTable,charNmbr)

local c;

return ClassFunction(charTable,List([1..Length(charTable.irreducibles)],

c->IrrCodim(charTable,charNmbr,c)));

end;

70

##########################################################################

# CODIM PARTITION

# Input: CHARTABLE - character table for a group

# CHARNMBR - position of an irreducible character in

# charTable.irreducibles

#

# Output: a list P with each entry a list of conjugacy class numbers

# P[1] holds the class numbers of elts g with codim(V^g)=0

# P[2] holds the class numbers of elts g with codim(V^g)=1

# :

# P[dimV+1] holds the class numbers of elts with codim(V^g)=dimV

#

# WARNING: THIS FUNCTION ASSUMES THE IDENTITY IS IN THE FIRST COLUMN

# OF THE CHARTABLE

#

# THE FUNCTIONS IsMinimalInP AND MinimalClassFunction DEPEND

# ON THE OUTPUT OF THIS FUNCTION BEING A LIST AS DESCRIBED ABOVE

#-------------------------------------------------------------------------

CodimPartition:=function(charTable,charNmbr)

local P, P_i, dimChar, classNmbr;

# get the dimension of the representation

dimChar:=Eigenvalues(charTable,charTable.irreducibles[charNmbr],1)[1];

# make a list with dimV+1 slots (one for each possible codimension),

# each slot holding an empty list []

P:=List([1..(dimChar+1)],P_i->[]);

# for each conjugacy class number, get codim(g) for elts g in

# that class, and store the class number in the corresponding slot

# (note: the numbering is offset by 1 since GAP labels slots in a list

# starting with 1 instead of 0)

for classNmbr in [1..Length(charTable.irreducibles)] do

Add( P[IrrCodim(charTable,charNmbr,classNmbr)+1], classNmbr);

od;

return P;

end;

71

##########################################################################

# IS MINIMAL IN P

# Input: CHARTABLE - character table of a reflection group

# CHARNMBR - position of a reflection representation in

# CHARTABLE.irreducibles

# CLASSK - position of a conjugacy class in CHARTABLE.classes

# P - partition of conjugacy classes according to codimension

#

# NOTE: This function assumes that P is a list of length dimV+1 with

# P[0+1]=list of positions of classes with codimension 0

# P[1+1]=list of positions of classes with codimension 1

# :

# P[n+1]=list of positions of classes with codimension n=dimV

#-------------------------------------------------------------------------

IsMinimalInP:=function(charTable,charNmbr,classk,P)

local d, smallercount, flag, i, classi, classj;

d:=IrrCodim(charTable,charNmbr,classk);

# if class consists of reflections, it is minimal

if d=1 then

return true;

fi;

# if class consists of elts acting as identity, it is not minimal

if d=0 then

return false;

fi;

# set up a counter to see if there are any classes below classk

# in the poset

smallercount:=0;

# compute ClassAlgConst(classi,classj,classk) for nontrivial

# classes classi and classj having codimensions adding to codim(classk)

for i in [1..(d-1)] do

for classi in P[i+1] do

for classj in P[(d-i)+1] do

72

smallercount:=smallercount

+ClassMultCoeffCharTable(charTable,classi,classj,classk);

od;

od;

od;

if smallercount > 0 then

return false;

fi;

return true;

end;

##########################################################################

# MINIMAL CLASS FUNCTION

# Input: CHARTABLE - character table of a reflection group

# CHARNMBR - position of a reflection representation in

# CHARTABLE.irreducibles

#

# Output: a class function with g->1 if g is minimal in the codim poset

# g->0 otherwise

#-------------------------------------------------------------------------

MinimalClassFunction:=function(charTable,charNmbr)

local P, minClassList, i;

P:=CodimPartition(charTable,charNmbr);

minClassList:=List([1..Length(charTable.irreducibles)],i->0);

for i in [1..Length(charTable.irreducibles)] do

if IsMinimalInP(charTable,charNmbr,i,P) then

minClassList[i]:=1;

fi;

od;

73

return ClassFunction(charTable,minClassList);

end;

##########################################################################

# LENGTH PARTITION

# Input: CHARTABLE - character table of a group G

# GENCLASSNUMS - conjugacy class numbers for a set of generators

# of the group G

# (ASSUMING 1 is not in GENCLASSNUMS)

#

# Output: a partition of the conjugacy classes of group G according to

# length with respect to the conjugacy classes given in

# GENCLASSNUMS

#

# Definition of length(g):

# length(g)=min number of elts needed to express g as a product

# of elts from the conjugacy classes in GENCLASSNUMS

# This length function is constant on conjugacy classes.

#

# The partition is returned as a list. Each element of the list

# is itself a list: the first entry is a length (possibly

# "infinity"); the second entry is a list of class numbers of

# elts with that length.

#-------------------------------------------------------------------------

LengthPartition:=function(charTable, genClassNums)

local P, G, classes, remainingClasses, i, classi, classj, classk,

previouslength, currentlength;

G:=charTable.group;

classes:=ConjugacyClasses(G);

# P will be the partition:

# [0,[1]] says the identity has length zero

# [1, genClassNums] says the generating classes have length 1

# IT IS ASSUMED THAT GENCLASSNUMS DOES NOT CONTAIN THE IDENTITY

P:=[ [0,[1]], [1,genClassNums] ];

# list all the class numbers [1,2,3,...]

remainingClasses:=Set(List([1..Length(classes)],i->i));

# remove 1 and all the genClassNums from remainingClasses

# (since their length is already known)

# note: SubtractSet is destructive

SubtractSet(remainingClasses,P[1][2]);

74

SubtractSet(remainingClasses,P[2][2]);

previouslength:=1;

currentlength:=2;

while Length(P[previouslength+1][2])>0 and

Size(remainingClasses)>0 do

# There were some classes with previouslength, so now we need

# to look for classes of currentlength, which will be stored in

# slot currentlength+1 of P

Add(P,[currentlength,[]]);

for classi in genClassNums do

for classj in P[previouslength+1][2] do

for classk in remainingClasses do

if (not classk in P[currentlength+1][2]) and

ClassMultCoeffCharTable(charTable,

classi,classj,classk)>0 then

# can multiply an element from generating class with

# an element with previouslength and get an element

# in classk, so classk has currentlength

AddSet(P[currentlength+1][2],classk);

fi;

od;

od;

od;

SubtractSet(remainingClasses,P[currentlength+1][2]);

previouslength:=currentlength;

currentlength:=currentlength+1;

od;

if Length(P[previouslength+1][2])=0 and Size(remainingClasses)>0 then

75

P[previouslength+1][1]:="infinity";

AddSet(P[previouslength+1][2],remainingClasses);

fi;

return P;

end;

##########################################################################

# LENGTH CLASS FUNCTION

# Input: CHARTABLE - character table of a group

# GENCLASSNUMS - positions of conjugacy classes whose elements

# generate the group (the identity should NOT be

# in this list)

#

# Output: a class function with g->length(g)

# [length(g)=min number of elts needed to express g as a product

# of elts from the conjugacy classes in GENCLASSNUMS]

# if the classes specified in GENCLASSNUMS do not generate the

# group, an error message is printed and nothing is returned

#-------------------------------------------------------------------------

LengthClassFunction:=function(charTable,genClassNums)

local G, classes, lengthlist, i, P, Pd, c;

G:=charTable.group;

classes:=ConjugacyClasses(G);

# store lengths of elts in each conj class

lengthlist:=List([1..Length(classes)],i->-1);

P:=LengthPartition(charTable,genClassNums);

# each Pd in the length partition P is a list [d,[c_1,...,c_r]]

# -the first slot gives a length d

# -the second slot is the list of positions of the conjugacy classes

# having length d

# if a class is not generated, it will be in a slot ["infinity",[...]]

for Pd in P do

for c in Pd[2] do

lengthlist[c]:=Pd[1];

od;

76

od;

# < evaluates to true when objects of different types are compared

if Maximum(lengthlist) < "infinity" then

return ClassFunction(charTable,lengthlist);

fi;

Print("Some classes not generated. Could not make class function.\n");

end;

##########################################################################

# REFLECTION CLASSES

# Input: CHARTABLE - character table for a reflection group

# CHARNMBR - position of a reflection representation in the

# the list CHARTABLE.irreducibles

#

# Output: list of positions in CHARTABLE.classes corresponding to

# conjugacy classes of reflections

#-------------------------------------------------------------------------

RefClasses:=function(charTable,charNmbr)

return CodimPartition(charTable,charNmbr)[2];

end;

##########################################################################

# REFLECTION CLASS FUNCTION

# Input: CHARTABLE - character table of a reflection group

# CHARNMBR - position of a reflection representation in

# CHARTABLE.irreducibles

#

# Output: a class function with g->1 if g is a reflection

# g->0 if g is not a reflection

#-------------------------------------------------------------------------

RefClassFunction:=function(charTable,charNmbr)

local charfunctionrefs, refs, r, i;

charfunctionrefs:=List([1..Length(charTable.irreducibles)],i->0);

refs:=RefClasses(charTable,charNmbr);

77

for r in refs do

charfunctionrefs[r]:=1;

od;

return ClassFunction(charTable,charfunctionrefs);

end;

##########################################################################

# GET REF REP

# Input: CHARTABLE - character table of a reflection group

#

# Output: position of a reflection representation in the list

# CHARTABLE.irreducibles of irreducible characters of the

# reflection group

#-------------------------------------------------------------------------

GetRefRep:=function(charTable)

local classes, i, refrepCharValueList;

classes:=ConjugacyClasses(charTable.group);

refrepCharValueList:=List([1..Length(classes)],

i->ReflectionCharValue(charTable.group,classes[i].representative));

for i in [1..Length(charTable.irreducibles)] do

if ScalarProduct(ClassFunction(charTable,charTable.irreducibles[i]),

ClassFunction(charTable,refrepCharValueList))=1 then

return i;

fi;

od;

end;

##########################################################################

#EOF

78

79

APPENDIX B

POINCARÉ POLYNOMIALS FOR ABSOLUTE REFLECTION LENGTH AND

CODIMENSION

Tables B.1 and B.2 display, for each exceptional complex reflection group G4 − G37, a

two-variable polynomial whose xlyd term indicates the number of elements in the group

that have absolute reflection length l and codimension d. The polynomials for the Coxeter

groups G23, G28, G30, and G35 − G37 are included for completeness, though their elegant

factorization has long been known (see Shephard-Todd [21]). We computed the polynomials

for the non-Coxeter groups using the code in Appendix A.

G4 1 + 8xy + (14x2 + x3)y2

G5 1 + 16xy + (52x2 + 3x3)y2

G6 1 + 14xy + (31x2 + 2x3)y2

G7 1 + 22xy + (85x2 + 36x3)y2

G8 1 + 18xy + (69x2 + 8x3)y2

G9 1 + 30xy + (145x2 + 16x3)y2

G10 1 + 34xy + (211x2 + 42x3)y2

G11 1 + 46xy + (327x2 + 202x3)y2

G12 1 + 12xy + (23x2 + 12x3)y2

G13 1 + 18xy + (47x2 + 30x3)y2

G14 1 + 28xy + (101x2 + 14x3)y2

G15 1 + 34xy + (157x2 + 96x3)y2

G16 1 + 48xy + (448x2 + 102x3 + x4)y2

G17 1 + 78xy + (875x2 + 246x3)y2

G18 1 + 88xy + (1240x2 + 471x3)y2

G19 1 + 118xy + (1853x2 + 1628x3)y2

G20 1 + 40xy + (244x2 + 75x3)y2

G21 1 + 70xy + (505x2 + 144x3)y2

G22 1 + 30xy + (119x2 + 90x3)y2

Table B.1. Reflection Length and Codimension Polynomials for Groups G4 −G22

80

G23 (1 + xy)(1 + 5xy)(1 + 9xy)

G24 1 + 21xy + 119x2y2 + (147x3 + 48x4)y3

G25 1 + 24xy + (174x2 + 9x3)y2 + (368x3 + 72x4)y3

G26 1 + 33xy + (318x2 + 9x3)y2 + (812x3 + 123x4)y3

G27 1 + 45xy + 519x2y2 + (1033x3 + 560x4 + 2x5)y3

G28 (1 + xy)(1 + 5xy)(1 + 7xy)(1 + 11xy)

G29 1 + 40xy + 530x2y2 + (2600x3 + 120x4)y3 + (3187x4 + 1200x5 + 2x6)y4

G30 (1 + xy)(1 + 11xy)(1 + 19xy)(1 + 29xy)

G31 1 + 60xy + (1210x2 + 60x3)y2 + (9300x3 + 1800x4)y3 + (18747x4 + 13620x5 + 1282x6)y4

G32 1 + 80xy + (2220x2 + 90x3)y2 + (24800x3 + 3600x4)y3 + (89082x4 + 35646x5 + x6)y4

G33 1 + 45xy + 750x2y2 + (5670x3 + 80x4)y3 + (18609x4 + 1440x5)y4 + (18685x5 + 6480x6 + 80x7)y5

G34 1 + 126xy + 6195x2y2 + (149940x3 + 1120x4)y3 + (1834119x4 + 70560x5)y4

+(10121454x5 + 1428000x6 + 10080x7)y5 + (15821171x6 + 9243108x7 + 504912x8 + 252x9 + 2x10)y6

G35 (1 + xy)(1 + 4xy)(1 + 5xy)(1 + 7xy)(1 + 8xy)(1 + 11xy)

G36 (1 + xy)(1 + 5xy)(1 + 7xy)(1 + 9xy)(1 + 11xy)(1 + 13xy)(1 + 17xy)

G37 (1 + xy)(1 + 7xy)(1 + 11xy)(1 + 13xy)(1 + 17xy)(1 + 19xy)(1 + 23xy)(1 + 29xy)

Table B.2. Reflection Length and Codimension Polynomials for Groups

G23 −G37

81

82

APPENDIX C

HOCHSCHILD COHOMOLOGY TABLES FOR GROUP G6

In this appendix, we summarize the Hochschild cohomology of the skew group algebra

S(V )#G, where V is any irreducible representation of the reflection group G6 (as num-

bered in the Shephard-Todd classification). The character table as displayed by the software

GAP [19] is included here in order to give context for the numbering of the characters in the

cohomology tables. The cohomology tables are explained in Example 4.2.1.

#########################################################

# CHARACTER TABLE OF REFLECTION GROUP G6

gap> DisplayCharTable(t6);

G6

2 4 3 4 2 2 2 2 2 2 2 2 3 4 4

3 1 . 1 1 1 1 1 1 1 1 1 . 1 1

1a 2a 2b 3a 12a 12b 6a 3b 12c 12d 6b 4a 4b 4c

2P 1a 1a 1a 3b 6b 6b 3b 3a 6a 6a 3a 2b 2b 2b

3P 1a 2a 2b 1a 4b 4c 2b 1a 4c 4b 2b 4a 4c 4b

5P 1a 2a 2b 3b 12d 12c 6b 3a 12b 12a 6a 4a 4b 4c

7P 1a 2a 2b 3a 12b 12a 6a 3b 12d 12c 6b 4a 4c 4b

11P 1a 2a 2b 3b 12c 12d 6b 3a 12a 12b 6a 4a 4c 4b

X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

X.2 1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1

X.3 1 -1 1 A -A -A A /A -/A -/A /A 1 -1 -1

X.4 1 -1 1 /A -/A -/A /A A -A -A A 1 -1 -1

X.5 1 1 1 A A A A /A /A /A /A 1 1 1

X.6 1 1 1 /A /A /A /A A A A A 1 1 1

X.7 2 . -2 -A B -B A -/A /B -/B /A . D -D

X.8 2 . -2 -A -B B A -/A -/B /B /A . -D D

X.9 2 . -2 -1 C -C 1 -1 -C C 1 . D -D

X.10 2 . -2 -1 -C C 1 -1 C -C 1 . -D D

X.11 2 . -2 -/A -/B /B /A -A -B B A . D -D

X.12 2 . -2 -/A /B -/B /A -A B -B A . -D D

X.13 3 -1 3 . . . . . . . . -1 3 3

X.14 3 1 3 . . . . . . . . -1 -3 -3

A = E(3)^2 = (-1-ER(-3))/2 = -1-b3

B = E(12)^11

C = E(4) = ER(-1) = i

D = 2*E(4) = 2ER(-1) = 2i

83

###########################################################

# HOCHSCHILD COHOMOLOGY TABLES FOR GROUP G6

#

# NOTE: det(g) line displays 1 if det(g)=1

# . otherwise

___________________________________________________________

G6 with 2-dimensional rep X.7,8,11,12

2 4 3 4 2 2 2 2 2 2 2 2 3 4 4

3 1 . 1 1 1 1 1 1 1 1 1 . 1 1

1a 2a 2b 3a 12a 12b 6a 3b 12c 12d 6b 4a 4b 4c

HH^0(g) * 0 0 0 0 0 0 0 0 0 0 0 0 0

HH^1(g) * 0 0 0 0 0 0 0 0 0 0 0 0 0

HH^2(g) * 0 0 0 0 0 0 0 0 0 0 0 0 0

* = (S(V) tensor \bigwedge^k(V^*))^G

det(g) 1 . 1 . . . . . . . . 1 . .

codim(g) 0 1 2 1 2 2 2 1 2 2 2 2 2 2

___________________________________________________________

G6 with 2-dimensional rep X.9,10

2 4 3 4 2 2 2 2 2 2 2 2 3 4 4

3 1 . 1 1 1 1 1 1 1 1 1 . 1 1

1a 2a 2b 3a 12a 12b 6a 3b 12c 12d 6b 4a 4b 4c

HH^0(g) * 0 0 0 0 0 0 0 0 0 0 0 0 0

HH^1(g) * 0 0 0 0 0 0 0 0 0 0 0 0 0

HH^2(g) * 0 0 0 0 0 0 0 0 0 0 0 0 0

* = (S(V) tensor \bigwedge^k(V^*))^G

det(g) 1 . 1 1 . . 1 1 . . 1 1 . .

codim(g) 0 1 2 2 2 2 2 2 2 2 2 2 2 2

84

___________________________________________________________

G6 with 3-dimensional rep X.13

2 4 3 4 2 2 2 2 2 2 2 2 3 4 4

3 1 . 1 1 1 1 1 1 1 1 1 . 1 1

1a 2a 2b 3a 12a 12b 6a 3b 12c 12d 6b 4a 4b 4c

HH^0(g) * 0 * 0 0 0 0 0 0 0 0 0 * *

HH^1(g) * 0 * 0 0 0 0 0 0 0 0 0 * *

HH^2(g) * 2m1 * 2m1 2m1 2m1 2m1 2m1 2m1 2m1 2m1 2m1 * *

HH^3(g) * 3m1 * 3m1 3m1 3m1 3m1 3m1 3m1 3m1 3m1 3m1 * *

* = (S(V) tensor \bigwedge^k(V^*))^G

2m1 = S(V^g) tensor C

[require poly part to have degree 2 mod 1]

3m1 = S(V^g) tensor (V^g)^*

[require poly part to have degree 3 mod 1]

det(g) 1 1 1 1 1 1 1 1 1 1 1 1 1 1

codim(g) 0 2 0 2 2 2 2 2 2 2 2 2 0 0

___________________________________________________________

G6 with 3-dimensional rep X.14

2 4 3 4 2 2 2 2 2 2 2 2 3 4 4

3 1 . 1 1 1 1 1 1 1 1 1 . 1 1

1a 2a 2b 3a 12a 12b 6a 3b 12c 12d 6b 4a 4b 4c

HH^0(g) * 0 * 0 0 0 0 0 0 0 0 0 0 0

HH^1(g) * 0 * 0 0 0 0 0 0 0 0 0 0 0

HH^2(g) * 0 * 2m2 0 0 2m2 2m2 0 0 2m2 2m2 0 0

HH^3(g) * 0 * 3m2 0 0 3m2 3m2 0 0 3m2 3m2 0 0

* = (S(V) tensor \bigwedge^k(V^*))^G

2m2 \subset S(V^g) tensor C

[require poly part to have degree 2 mod 2]

3m2 \subset S(V^g) tensor (V^g)^*

[require poly part to have degree 3 mod 2]

det(g) 1 . 1 1 . . 1 1 . . 1 1 . .

codim(g) 0 1 0 2 3 3 2 2 3 3 2 2 3 3

############################################################

85

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